UNIT - 5TRIVANDRUM: T.C.No: 5/1703/30, Golf Links Road, Kowdiar Gardens, H.B. Colony, TVM, 0471-2438271KOCHI: Bldg.No.41/352, Mulloth Ambady Lane, Chittoor Road, Kochi - 11, Ph: 0484-2370094Today’s Mathiit’ians..... Tomorrow’s IITi’ians.....CONTENTS* SynopsisQuestions* Level - 1* Level - 2* Level - 3Answers* Level - 1* Level - 2* Level - 3e- Learni ng Resourcesw w w . m a t h i i t . i nDefinite IntegrationMaterialSYNOPSISDefinite IntegralMathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in1.ab∫f(x) dx = F(b) - F(a) where ∫f(x) dx = F(x) + c2. If ab∫f(x) dx = 0 ⇒ then the equation f(x) = 0has atleast one root lying in (a, b) providedf is a continuous function in (a, b) .3. PROPERTIES OF DEFINITE INTEGRAL :P - 1ab∫f(x) dx = ab∫f(t) dtP - 2ab∫f(x) dx = -ba∫f(x) dxP - 3 a

b∫f(x) dx = ac∫f(x) dx + cb∫f(x) dx , where cmay lie inside or outside the interval [a, b] . Thisproperty is to be used when f is piecewisecontinuous in (a, b) .P - 4−∫aaf(x) dx = 0 if f(x) is an oddfunction i.e. f(x) = � f(�x) .= 20a∫f(x) dx if f(x) is an even functioni.e. f(x) = f(�x) .P � 5ab∫f(x) dx = ab∫f(a + b � x) dx,In particular 0a∫f(x) dx = 0a∫f(a � x)dxP � 602a∫f(x) dx = 0a∫f(x) dx +

0a∫f(2a � x) dx= 2 0a∫f(x) dx if f(2a � x) = f(x)= 0 if f(2a � x) = � f(x)P � 70na∫f(x) dx = n0a∫f(x) dx ; where‘a’is theperiod of the function i.e. f(a + x) = f(x)P � 8a nTb nT++∫ f(x) dx = ab∫f(x) dx where f(x)is periodic with period T & n ∈ I .P � 9mana∫f(x) dx = (n � m) 0a∫f(x) dx if f(x)is periodic with period �a� .P �

10 If f(x) ≤ φ (x) �or a ≤ x ≤ b thenab∫�(x) dx ≤ ab∫ φ (x) dxP

- 11 ∫ab� x d x ( ) ≤ ab∫ |�(x) | dx .P - 12 I� �(x) ≥ 0 on the interval [a, b], thenab∫ �(x) dx ≥ 0 .4. WALLI’S FORMULA :02 π/∫sinnx . cosmx dx=[ ] [ ] ( ) ( ) ( ).... ( ) ( )....( ) ( ) ( )....n n n or m m orm n m n m n or− − − − −+ + − + −1 3 5 1 2 1 3 1 22 4 1 2Kwhere K = 2π i� both m & n are even(m, n ∈ N) = 1 otherwise5. I� h(x) & g(x) are di��erentiable �unctions o� x then,dd x g xh x( )( )∫ �(t) dt = � [h (x)] . ) x ( h′ - � [g (x)] . ) x ( g′6. Limitn→∞1n|.

`

, ∑·−rn11 f rn|.

�, = 01∫f(x) dx .7. For a monotonic decreasing function in (a, b) ;f(b).(b � a) < ab∫ f(x) dx < f(a).(b � a)8. For a monotonic increasing function in (a, b) ;f(a).(b � a) < ab∫f(x) dx < f(b).(b � a)DEFINITIONS AND RESULTSw w w. m a t h i i t . i n LEVEL � 1 (Fundamentals of Definite Integration)Mathiit e� Learni ng Res ources doubt@mathiit.in www.mathiit.in1. 1210nxe dx ·∫ (a) 0 (b) 12(c) 13(d) 14.2. / 420tan xdxπ

·∫(a) 14π− (b) 14π+ (c) 14π− (d) 4π.3. / 20sin1 cosx xdxxπ +·+∫(a) log 2e− (b) log 2e (c) 2π(d) 0.4. / 20sinxe xdxπ·∫(a) ( )/ 2112 eπ− (b) ( )/ 2112 eπ

+ (c) ( )/ 2112 eπ−(d) ( )/ 22 1 eπ+ .5.2211 1xe dxx x| `− · . ,∫(a) 22ee + (b) 22ee − (c) 22ee −(d) None of these.6.( )( )/ 20cos1 sin 2 sinxdxx xπ·+ +∫(a) 4log3

(b) 1log3 (c) 3log4(d) None o� these.7.( )/ 25/ 321 cos1 cosxdxxππ+·−∫(a) 52 (b) 32 (c) 12(d) 25.8.12211xe dxx−·∫(a) 1 e + (b) 1 e − (c) 1 ee+(d) 1 ee

−.Mathiit e� Learni ng Res ources doubt@mathiit.in www.mathiit.in9. 11202sin1xdxx− | �· +. ,∫(a) 2log 22π− (b) 2log 22π+ (c) log 24π−(d) log 24π+.10. The value o� ( )232ax bx c− + +∫ depends on(a) The value of a (b) The value of b (c) The value of c (d) The value of a and b.11. / 4/ 6cos 2 ec x dxππ

·∫(a) log 3 (b) log 3 (c) log9 (d) None o� these.12. logbaxdxx ·∫(a) logloglogba| ` . , (b) ( ) log log baba| ` . , (c) ( )1log log2baba| ` . , (d) ( )1log log2aabb| ` . ,13. 110tan x dx−·∫

(a) 1log 24 2π− (b) 1log 22π − (c) log 24π− (d) log 2 π − .14.( ) ( )1201dxax b x·+ −∫(a) ab (b) ba (c) ab (d) 1ab.15. / 22/ 4cos cos ec dππ θ θ θ ·∫(a) 2 1 − (b) 1 2 − (c) 2 1 +(d) None of these .16.( )11/ 23/ 20 2sin1xdxx−·

−∫(a) 1log 24 2 eπ+ (b) 1log 24 2 eπ− (c) log 22 eπ+(d) log 22 eπ−.17. / 202 cosdxxπ·+∫(a) 11 1tan3 3− | � . , (b) ( )13 tan 3− (c) 12 1tan3 3− | �

. , (d) ( )12 3 tan 3−.Mathiit e� Learni ng Res ources doubt@mathiit.in www.mathiit.in18.1120tan1xdxx−·+∫(a) 28π (b) 216π (c) 24π(d) 232π.19. The value o� integral ( )2/21/sin 1/ xdxxππ ·∫(a) 2 (b) -1 (c) 0 (d) 1.20.2/ 20.sin2 4x

xe dxπ π | `+ · . ,∫(a) 1 (b) 2 2 (c) 0 (d) None o� these.21. 101xxedxe−− ·+∫(a) 1 1log 1ee e+ | `− + . , (b) 1 1log 12ee e+ | �− + . , (c) 1 1log 12ee e+ | �+ − . , (d) None of these.22. / 40sin cos9 16sin 2x xdx

xπ +·+∫(a) 1log320 (b) log3 (c) 1log520(d) None o� these.23. ( )/ 2/ 4logsin cotxe x x dxππ + ·∫(a) / 4log 2 eπ (b) / 4log2 eπ− (c) / 41log 22eπ(d) / 41log 22eπ− .24.11/ 22 0sin1x xdxx−·

−∫(a) 1 32 12π+ (b) 1 32 12π− (c) 1 32 12π− (d) None of these.25. 2022xdxx+·−∫(a) 2 π + (b) 32π + (c) 1 π + (d) None o� these.26.01 sindxxπ·+∫(a) 0 (b) 12 (c) 2 (d) 32.27. 201 sin2xdxπ+ ·∫(a) 0 (b) 2 (c) 8 (d) 4.28. 1

10cos x dx−·∫(a) 0 (b) 1 (c) 2 (d) None o� these.29. / 20cos1 cos sinxdxx xπ·+ +∫(a) 1log 24 2π+ (b) log 24π+ (c) 1log 24 2π−(d) log 24π−.30. ( )/ 6202 3 cos3 x xdxπ+ ·∫(a) ( )11636 π + (b) ( )11636

π − (c) ( )211636 π −(d) ( )211636 π +.31. / 240sin cos1 sinx xdxxπ·+∫(a) 2π (b) 4π (c) 6π(d) 8π.32. / 46 20tan sec x x dxπ·∫(a) 17 (b) 27 (c) 1(d) None o� these.33. / 63

0sincosxdxxπ·∫(a) 23 (b) 16 (c) 2 (d) 13.34. / 220sin coscos 3cos 2x x dxdxx xπ·+ +∫(a) 8log9| ` . , (b) 9log8| ` . , (c) ( ) log 8 9 × (d) None o� these.35. The value o� the de�inite integral 12002 cos 1dx�orx x α πα < <+ +

∫ is equ�l to(�) sinα (b) ( )1t�n sinα− (c) sin α α (d) ( ) 1sin2αα −M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in36. The integr�l 231 1211t�n t�n1x xdxx x− −−| ` ++ · +. ,∫(a) π (b) 2π (c) 3π (d) None o� these.37. I� 221 21,logxeedx eI and I dxx x· ·∫ ∫then(a) 1 2I I · (b) 1 2I I > (c) 1 2I I < (d) None o� these.

38. / 2/ 4sinxe xdxππ−− ·∫(a) / 212e π −− (b) / 422 e π −− (c) ( )/ 4 / 42 e eπ π − −− + (d) 0.39.( )/ 2201 2cos2 cosxdxxπ +·+∫(a) 2π (b) π (c) 12(d) None o� these.40. 201 2 cosdxa x aπ

·− +∫(�) ( )22 1 �π− (b) ( )21 � π − (c) 21 �π−(d) None of these.41. ( )1 901 x dx − ·∫(a) 1 (b) 110 (c) 1110(d) 2.42. / 30cos3xdxπ·∫(a) π (b) 0 (c) 2π(d) 4π.43. The value o� / 401 tan1 tanxdxxπ +−∫ is(�)

1log 22− (b) 1log 24 (c) 1log 23(d) None of these.44. The v�lue of 10 x xdxe e−+∫ is(�) 1 1t�n1ee− − | ` +. , (b) 1 1t�n1ee− − | ` +. , (c) 4π(d) 1t�n4e π−+ .M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in45.1

1 loge xdxx+·∫(a) 32 (b) 12 (c) 1e(d) None o� these.46. I� 102log 1 log ,2 3xx dx a b| ` | `+ · + . , . ,∫ then(a) 3 3,2 2a b · · (b) 3 3,4 4a b · · − (c) 3 3,4 2� b · · (d) a b · .47.101dxx x ·+ −∫(�) 2 23 (b) 4 23

(c) 8 23(d) None of these.48. / 44 404sin 2sin cosdπ θ θθ θ ·+∫(a) / 4 π (b) / 2 π (c) π (d) None o� these.49.( )( )13011xe xdxx−·+∫(a) 4e (b) 14e− (c) 14e+(d) None of these.50. If ( ) ( )41 1, x x x φ + · then ( )21x dx φ ·∫(a)

1 32log4 17 (b) 1 32log2 17 (c) 1 16log4 17(d) None o� these.51.1/ 22 1/ 4dxx x·−∫(�) π (b) 2π (c) 3π(d) 6π.52. The v�lue of 203 xdxx∫ is(�) ( )223 1log3 − (b) 0 (c) 2 2log 3(d) 232.53. ( )20sin cos x x dxπ+ ·∫

(a) 0 (b) 2 (c) -2 (d) 1.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in54. / 420sec1 2sinxxπ+∫ is e�ual to(a) ( )1log 2 13 2 2π ]+ + ] ](b) ( )1log 2 13 2 2π ]+ − ] ](c) ( )3 log 2 12 2π ]+ − ] ](d) ( )3 log 2 12 2π ]+ + ] ]55. The v�lue of / 220sin1 cosxdxxπ+∫ is(�) / 2 π (b) / 4 π (c) / 3 π (d) / 6 π .

56. The v�lue of 21log x dx∫ is(�) log 2 / e (b) log 4 (c) log 4 / e (d) log 2 .57. The v�lue of 25234xdxx −∫ is(�) 152 log7e| `− . , (b) 152 log7e| `+ . ,(c) 2 4log 3 4log 7 4log 5e e e+ − + (d) 1 152 t�n7− | `− . ,.58. The v�lue of 2 2sin cos1 10 0sin cosx xt dt t dt− −+∫ ∫

(�) 2π (b) 1 (c) 4π(d) None of these.59. If for non�zero x , ( ) 1 15, �f x bfx x| `+ · − . , where , � b ≠ then ( )21f x dx ·∫(a) ( )2 21 7log2 52a a ba b ]− + ]+ ](b) ( )2 21 7log 2 52� � b� b ]− + ]− ](c) ( )2 21 7log 2 52� � b� b ]− − ]− ](d) ( )2 2

1 7log 2 52� � b� b ]− − ]+ ].60. If / 40t�n ,nnI dπθ θ ·∫ then 8 6I I + e�uals(a) 14 (b) 15 (c) 16(d) 17.61. 2/ 3204 9dxx ·+∫(a) 12π (b) 24π (c) 4π(d) 0.62. The value o� 412

011xdxx++∫ is(a) ( )13 46 π − (b) ( )13 46 π − (c) ( )13 46 π +(d) ( )13 46 π +.M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.inM�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in63. 2 30sin�

x x dx∫ equ�ls(�) ( )31 cos � − (b) ( )33 1 cos � − (c) ( )311 cos3 � − − (d) ( )311 cos3 � −64.

( )/ 40t�n cot x x dxπ+∫ equ�ls(�) 2π (b) 2π (c) 2π(d) 2π.65. 1011xdxx−+∫ equ�ls(�) 12π | `− . , (b) 12π | `+ . , (c) 2π(d) ( ) 1 π + .66.11edxx∫ is equ�l to(�) ∞ (b) 0 (c) 1 (d) ( ) log 1 e + .67.2

1logx xdxx ·∫(a) ( )2log x (b) ( )2 1log2 x (c) 2log2x(d) None o� these.68. / 22 2 2 20cos sindxa x b xπ·+∫(a) ab π (b) 2ab π (c) abπ(d) 2abπ.69. ( ) ( ) ( )/ 4 5 / 4 / 40 / 4 2cos sin sin cos cos sin x x dx x x dx x x dxπ π ππ π− + − + −∫ ∫ ∫ is equ�l to(�) 2 2 − (b) 2 2 2 − (c) 3 2 2 −(d) 4 2 2 −.70. ( ) ( )0

4 ,�

x dx � ≤ +∫ then(�) 0 4 � ≤ ≤ (b) 2 4 � − ≤ ≤ (c) 2 0 � − ≤ ≤ (d) 2 4 � or � ≤71. 0212 2dxx x− ·+ +∫(a) 0 (b) / 4 π (c) / 2 π (d) / 4 π − .72. 32111 dxx +∫ is equ�l to(�) /12 π (b) / 6 π (c) / 4 π (d) / 3 π .73. 31( 1)( 2)( 3) x x x dx − − − ·∫(a) 3 (b) 2 (c) 1 (d) 0.74. 322dxx x ·−∫(�) ( ) log 2/ 3 (b) ( ) log 1/ 4 (c) ( ) log 4/ 3 (d) ( ) log 8/ 3 .75.( )1583 1dxx x ·− +∫(�) 1 5log2 3

(b) 1 5log3 3 (c) 1 3log2 5(d) 1 3log5 5.76. The v�lue of 30sin dπθ θ∫ is(a) 0 (b) 3/ 8 (c) 4/ 3(d) π .77.1101sin 2tan1xdxx−| `+· −. ,∫(�) / 6 π(b) / 4 π (c) / 2 π(d) π .78. 3203 19xdx

x+·+∫(a) ( )log 2 212π+ (b) ( )log 2 22π+ (c) ( )log 2 26π+ (d) ( )log 2 23π+79. The value o� ( )2411dxx x +∫ is(a) 1 17log4 32 (b) 1 17log4 2 (c) 17log2(d) 1 32log4 17.80. The value o� ( )32211

xdxx x+−∫ is(�) 12log 26− (b) 16 1log9 6− (c) 4 1log3 6−(d) 16 1log9 6+ .81. The v�lue of 1logex dx∫ is(�) 0 (b) 1 (c) 1 e − (d) 1 e + .82. The v�lue of ( )2/ 20sin cos1 sin 2x xI dxxπ +·+∫ is(a) 3 (b) 1 (c) 2 (d) 0.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in83. / 830cos 4 dπθ θ ·∫(a) 2

3 (b) 14 (c) 13(d) 16.84. ( )832 31xdxx x−+∫ is equ�l to(�) ( )32log 3/ 2e (b) ( )3log 3/ e (c) ( )34log 3/ e (d) None of these85. The v�lue of 120xx e dx∫is equ�l to(�) 2 e − (b) 2 e + (c) 22 e − (d) 2e .86. Let 2 21 22 1 11dx dxI �nd Ixx· ·+∫ ∫ then

(a) 1 2I I > (b) 2 1I I > (c) 1 2I I · (d) 1 221 I > .87. The value o� ( )tan cot2 21/ /1 1x xe I et dt dtt t t+ ·+ +∫ ∫(a) -1 (b) 1 (c) 0 (d) None o� these.88. 3 / 4/ 4 1 cosdxxππ +∫ is e�ual to(a) 2 (b) -2 (c) 12(d) 12− .89. The v�lue of ( )2211e dxx ln x +∫ is(�) 2/ 3 (b) 1/ 3 (c) 3/ 2(d) ln 2.90. / 2

2/ 4cosec xdxππ ·∫(a) -1 (b) 1 (c) 0 (d) 12.91. I� ( )1/ 2log2,61xudueπ·−∫ then xe ·(a) 1 (b) 2 (c) 4 (d) -1.92. I� ( ) ( ) 1 2 , g g · then ( ) ( ) ( ) ( )2 11� ( ) � �g x � g x g x dx−∫ is equ�l to(�) 1 (b) 2 (c) 0 (d) None of these.M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.inM�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.inw w w. m � t h i i t . i n LEVEL � 2 (Pro�erties of Definite Integr�tion)1. ( )0sin xf x dxπ·∫(a) ( )0sin � x dxππ∫ (b) ( )0sin2 � x dxπ

π∫ (c) ( )/ 20sin2 � x dxππ∫ (d) None o� these.2./ 20cotcot tanxdxx xπ·+∫(a) π (b) 2π (c) 4π(d) 3π.3. / 201 tandπ θθ ·+∫(a) π (b) 2π (c) 3π(d) 4π.4. I� ( ) 3,xt

a� x t e dt ·∫ then ( )d� xdx ·(a) ( )3 23xe x x + (b) 3 xx e (c) 3 aa e (d) None o� these.5. 11x x dx− ·∫(a) 1 (b) 0 (c) 2 (d) -2.6. / 20log tan x dxπ·∫(a) log 22 eπ (b) log 22 eπ− (c) log 2eπ (d) 0.7. / 20logsin x dxπ·∫(a) log 22π | `− . ,

(b) 1log2π (c) 1log2π − (d) log22π.8. / 20cos sin1 sin cosx xdxx xπ −·+∫(a) 2 (b) -2 (c) 0 (d) None o� these.9. 112log2xdxx−− | `· +. ,∫(a) -2 (b)1 (c) -1 (d) 0.10. 117 41cos x xdx− ·∫(a) -2 (b)-1 (c) 0 (d) 2.11.3/ 2/ 23/ 2 3/ 20sincos sin

xdxx xπ·+∫(a) 0 (b) π (c) / 2 π(d) / 4 π.12. ( )/ 40log 1 tan dπθ θ + ·∫(a) log 24π (b) 1log4 2π (c) log 28π(d) 1log8 2π.13. 20sin 2cos da bπ θθθ ·−∫(�) 1 (b) 2 (c) 4π(d) 0.14. ( )101 f x dx −∫

h�s the s�me v�lue �s the integr�l(�) ( )10f x dx∫ (b) ( )10f x dx −∫ (c) ( )101 f x dx −∫(d) ( )11f x dx−∫.15. ( )1/ 21/ 21cos log1xx dxx− − ] | `· ]+. , ]∫(a) 0 (b) 1 (c) 1/ 2e(d) 1/ 22e .16. The value o� 12 01dxx x + −∫ is(�) 3π (b) 2π (c)

12(d) 4π.17. If ( )110, f x− ·∫ then(a) ( ) ( ) � x � x · − (b) ( ) ( ) f x f x − · − (c) ( ) ( ) 2 f x f x · (d) None o� these.18. 111 x dx− − ·∫(a) -2 (b) 0 (c) 2 (d) 4.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.inMathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in19. 30sin x x dxπ·∫(a) 43π (b) 23π (c) 0 (d) None o� these.20. 2221 x dx− − ·∫(a) 2 (b) 4 (c) 6 (d) 8.21./ 20cossin cosxdxx xπ

·+∫(a) 2 (b) 2π (c) 4π(d) None o� these.22. / 24 40sin coscos sinx x xdxx xπ·+∫(a) 0 (b) 8π (c) 28π(d) 216π.23. The correct evaluation o� / 20sin4x dxπ π | `− . ,∫ is(�) 2 2 + (b) 2 2 − (c) 2 2 − + (d) 0.24. ( )0�

f x dx ·∫(a) ( )0a� a x dx +∫ (b) ( )

02a� a x dx +∫ (c) ( )0a� x a dx −∫(d) ( )0�

f � x dx −∫.25. / 20sin cos x x dxπ− ·∫(a) 0 (b) ( )2 2 1 − (c) 2 1 −(d) ( )2 2 1 + .26.0cos x dxπ·∫(a) π (b) 0 (c) 2 (d) 1.27. The value o� the integral / 44/ 4sin x dxππ−− ·∫(a) 3/2 (b) -8/3 (c) 3/8 (d) 8/3.28. 1.520, x dx ] ] ∫ where [ . ] denotes the greatest integer �unction, e�uals(a) 2 2 + (b) 2 2 − (c) 2 2 − + (d) 2 2 − − .29.0t�n

sec t�nx xdxx xπ·+∫(a) 12π− (b) 12ππ | `+ . , (c) 12π+(d) 12ππ | `− . ,.30.0t�nsec cosx xdxx xπ·+∫(a) 24π (b) 22π (c) 232

π(d) 23π.31. 13 21sin cos x xdx− ·∫(a) 0 (b) 1 (c) 12(d) 2.32. For any integer n, the integral ( )2sin 30cos 2 1xe n x dxπ+ ·∫(a) -1 (b) 0 (c) 1 (d) π .33.1/logeex dx ·∫(a) 11e− (b) 12 1e| `− . , (c) 11 e−−(d) None of these.34. [ ] ( )/ 20sin x x dxπ

−∫is equ�l to (where [.] re�resents gre�test integer function)(�) 28π (b) 218π− (c) 228π− (d) None of these.35. The v�lue of the integr�l ( )101nI x x dx · −∫ is(�) 11 n + (b) 12 n + (c) 1 11 2 n n−+ +(d) 1 11 2 n n++ +.36. The v�lue of [ ]22sin , x dxππ∫ where [ . ] re�resents the gre�test integer function, is(�) π − (b) 2π − (c) 53π−(d) 5

3π.37. The v�lue of / 2301 t�ndxxπ+∫ is(�) 0 (b) 1 (c) 2π(d) 4π.38. The v�lue of 3 / 4/ 4,1 sin dππφφφ +∫ is(a) tan8ππ (b) log tan8π (c) tan8π(d) None o� these.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in39. I� ( ) ( ) ( ), ba� a b x � x then x � x dx + − · ·∫(a) ( )2baa b� b x dx

+−∫ (b) ( )2b�

� bf x dx+∫ (c) ( )2b�

b �f x dx−∫ (d) None of these.40.0sin x x dxπ·∫(a) π (b) 0 (c) 1 (d) 2π.41. I� ( ) ( )20 02 ,a a� x dx � x dx ·∫ ∫ then(a) ( ) ( ) 2 � a x � x − · − (b) ( ) ( ) 2 f � x f x − · (c) ( ) ( ) � a x � x − · − (d) ( ) ( ) f � x f x − · .42. I� / 4 / 42 20 0sin cos , I x dx and J x dx then Iπ π· · ·∫ ∫(a) 4 Jπ− (b) 2J (c) J(d) 2

J.43. The v�lue of ( )513 1 x x dx − + −∫ is(�) 10 (b) 56 (c) 21 (d) 12.44. The v�lue of 325xdxx x − +∫ is(�) 1 (b) 0 (c) �1 (d) 12.45. The v�lue of 2cos 50cos 3xe x dxπ∫ is(�) 1 (b) �1 (c) 0 (d) None of these.46./ 2011 t�n dxxπ·+∫(a) 2π (b) 4π (c) 6π(d) 1.47. The value o� 21

1sin3x xdxx−−−∫ is(�) 0 (b) 10sin23xdxx −∫ (c) 21023xdxx−−∫(d) 210sin23x xdxx−−∫.48. 1111sin xdx−∫ is equ�l to(�) 10 8 6 4 2. . . .11 9 7 5 3 (b)

10 8 6 4 2. . . . .11 9 7 5 3 2π (c) 1 (d) 0.M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in49. To find the numeric�l v�lue of ( )222, �x qx s dx− + +∫ it is necess�ry to know the v�lues of const�nts(�) � (b) q (c) s (d) � �nd s.50.111log1xdxx−+ | `· −. ,∫(�) 2 (b) 1 (c) 0 (d) π .51. / 2/ 2cos1 xxdxeππ − ·+∫(a) 1 (b) 0 (c) -1 (d) None o� these.52. I� [ ] x denotes the greatest integer less than or e�ual to x , then the value o� the integral [ ]220x x dx∫ e�uals(a) 5/3 (b) 7/3 (c) 8/3 (d) 4/3.53. 3

0cos x dxπ·∫(a) -1 (b) 0 (c) 1 (d) π .54. 20logsin x dxπ·∫(a) 12 log2eπ | ` . , (b) log 2eπ (c) 1log2 2eπ | ` . ,(d) None o� these.55. I� ( ) � x is an odd �unction o� x , then ( )22cos � x dxππ−∫ is equ�l to(�) 0 (b) ( )20cos f x dxπ∫ (c) ( )202 sin f x dxπ∫ (d) ( )0cos f x dxπ

∫.56. 20sin x dxπ∫ is equ�l to(�) π (b) 2π (c) 0 (d) None of these.57. / 20sinsin cosxdxx xπ+∫ equ�ls(�) 2π (b) 3π (c) 4π(d) 6π.58. 111t�n x x dx−−∫ equ�ls(�) 12π | `− . , (b) 12π | `+

. , (c) ( ) 1 π − (d) 0.M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in59. ( ) sin cos�

�

x f x dx− ·∫(a) ( )02 sin cosax� x dx∫ (b) 0 (c) 1 (d) None o� these.60. The value o� 230sin dπθ θ∫ is(a) 0 (b) 3 / 8 (c) 8 / 3 (d) π .61. 21x dx−∫(�) 5 / 2 (b) 1 / 2 (c) 3 / 2 (d) 7 / 2.62. 302 x dx −∫ equ�ls(�) 2 / 7 (b) 5 / 2 (c) 3 / 2 (d) �3 / 2.63. The v�lue of sin/ 2sin cos022 2xx x dxπ+∫ is(�) 4π (b) 2

π (c) π (d) 2π.64. The v�lue of 1203 1 x dx −∫is(�) 0 (b) 4/ 3 3 (c) 3 / 7 (d) 5 / 6.65. 2 / 2cos2/ 2sin1 cosxxe dxxππ−−+∫ is equ�l to(�) 12e− (b) 1 (c) 0 (d) None of these.66. ( ) ( ) 2 , f x f x · − then ( )1.50.5xf x dx∫ equ�ls(�) ( )10f x dx∫ (b) ( )1.50.5f x dx∫ (c) ( )1.50.52 f x dx∫(d) 0.67.22

2/ 202xxxedxe eππ | `− . ,+∫ is(�) / 4 π (b) / 2 π (c) 2/16 eπ(d) 2/ 4 eπ.68. If [ ] x denotes the gre�test integer less th�n or equ�l to x , then the v�lue of 513 x dx − ] ] ∫ is(�) 1 (b) 2 (c) 4 (d) 8.M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in69.22| | x dx− ·∫(a) 0 (b) 1 (c) 2 (d) 4.70. Su��ose � is such that ( ) ( ) � x � x − · − for every re�l x �nd ( ) ( )1 00 15, f x dx then f t dt−· ·∫ ∫(a) 10 (b) 5 (c) 0 (d) -5.71. Let ( ) ( )1 2sin , sin ,a aa a

I x� x dx I � x dxπ π − −· ·∫ ∫then 2I is e�ual to(a) 12 Iπ (b) 1I π (c) 12Iπ(d) 12I .72. 1/ 21/ 21cos .1xx ln dxx−+ | ` −. ,∫ is equ�l to(�) 0 (b) 1 (c) 2 (d) ln 3.73. The v�lue of 21logeeexdxx− ∫ is(�) 32 (b) 52 (c) 3 (d) 5.74. If ( ) ( )3

2sin , 2,2,cos xe x xf x then f x dxotherwise −� ≤· �¹ ∫ is e�ual to(a) 0 (b) 1 (c) 2 (d) 3.75. I� : : � R Rand g R R → → are one to one, real valued �unctions, then the value o� the integral( ) ( ) ( ) ( ) ( ) ( )� x � x g x g x dxππ − + − − −∫ is(�) 0 (b) π (c) 1 (d) None of these.76. / 3/ 61 cotdxxππ+∫ is(�) / 3 π (b) / 6 π (c) /12 π(d) / 2 π.77. The v�lue of 2/ 3/ 22/ 3 2/ 30sinsin cosxdxx xπ+∫ is

(�) / 4 π (b) / 2 π (c) 3 / 4 π(d) π .78. ( )121log 1 x x dx− + + ·∫(a) 0 (b) log 2 (c) 1log2(d) None o� these.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.inMathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in79. The value o� the integral ( ) ( )2cos sin , int ax bx dx a and bare egerππ − −∫ is(�) π − (b) 0 (c) π (d) 2π .80. 01 cos 22xdxπ +∫is equ�l to(�) 0 (b) 2 (c) 1 (d) �1.81. ( )20�

f x dx ·∫(a) ( )02 a� x dx∫ (b) 0 (c) ( ) ( )0 02a a

� x dx � a x dx + −∫ ∫ (d) ( ) ( )20 02� �f x dx f � x dx + −∫ ∫.82. 2sin 30cosxe x dxπ∫ is equ�l to(�) �1 (b) 0 (c) 1 (d) π .83. Find the v�lue of 902 , x dx ]+ ]∫where [ . ] is the gre�test integer function(�) 31 (b) 22 (c) 23 (d) None of these.84. The v�lue of 220, x dx ] ] ∫where [ . ] is the gre�test integer function(�) 2 2 − (b) 2 2 + (c) 2 1 − (d) 2 2 − .85. [ ]10000x xe dx−∫is(�) 10001 e − (b) 100011ee−− (c) ( ) 1000 1 e − (d)

11000e −.86. The v�lue of the ingr�l 11�nnnxdx� x x−− +∫ is(�) 2�

(b) 22n�n+ (c) 22n�n−(d) None of these.87. / 20sin 2 log t�n x x dxπ∫ is equ�l to(�) π (b) / 2 π (c) 0 (d) 2π.88. The integr�l [ ]1/ 21/ 21log1xx dxx−| + ` | `+ −. , . ,

∫ equ�l ( where [.] is the gre�test integer function )(�) 12− (b) 0 (c) 1 (d) 12log2.89. ( )20sin sin x x dxπ+ ·∫(a) 0 (b) 4 (c) 8 (d) 1.90. The value o� ( )/ 23/ 23sin sin x x dxππ − +∫ is(�) 3 (b) 2 (c) 0 (d) 103.91. The v�lue of 1012I x x dx · −∫ is(�) 1/3 (b) 1/4 (c) 1/8 (d) None of these.92. The v�lue of 805 x dx −∫(�) 17 (b) 12 (c) 9 (d) 18.93. 201 x dx − ·∫(a) 0 (b) 2 (c) 1/2 (d) 1.94. [ ]22x dx−

·∫ (where [.] denotes greatest integer �unction)(a) 1 (b) 2 (c) 3 (d) 4.95. 11201tan1 dxx x− | ` − +. ,∫(�) 2 ln (b) 2 ln − (c) 22 lnπ+(d) 22 lnπ−.96. The v�lue of , 0b�

xdx � bx < <∫ is(�) ( )� b − + (b) b � − (c) � b − (d) � b + .97. The v�lue of 2221 11 1x x�ln qln r dxx x−−| `+ − | ` | `+ +

− +. , . ,. ,∫de�ends on(�) The v�lue of � (b) The v�lue of q (c) The v�lue of r (d) The v�lue of � �nd q.98. 01 sinxdxxπ+∫ is equ�l to(�) π − (b)2π (c) π (d) None of these.M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.inM�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in99. The v�lue of 3221 x dx− −∫ is(�) 13 (b) 143 (c) 73(d) 283.100. If ( ) ( )201 , f x x then f x dx · −∫ is(�) 1 (b) 0 (c) 2 (d) �2.101. If ( ) ( )/ 20 0sin sin , xf x dx A f x dx then Aπ π·∫ ∫is(a)

2π (b) π (c) 4π(d) 0.102. ( ) ( )/ 20sin cos log sin cos x x x x dxπ− + ·∫(a) -1 (b) 1 (c) 0 (d) None o� these.103. The �unction ( )1x dtL xt·∫ satis�ies the e�uation(a) ( ) ( ) ( ) L x y L x L y + · + (b) ( ) ( )xL L x L yy| `· + . , (c) ( ) ( ) ( ) L xy L x L y · + (d) None o� these.104. The value o� integral 2 10xe dx∫lies in the interval(a) ( 0,1 ) (b) ( -1,0 ) (c) ( 1, e ) (d) None o� these.105. I� ( ) ( )32 20 0cos cos , P � x dx and Q � x dx thenπ π· ·∫ ∫(a) P - Q = 0 (b) P - 2Q = 0 (c) P - 3Q = 0 (d) P - 5Q = 0.106. Let a, b, c be non - zero real numbers such that( ) ( )3 32 20 13 2 3 2 , ax bx c dx ax bx c dx then + + · + +∫ ∫(a) a + b + c = 3 (b) a + b + c = 1 (c) a + b + c = 0 (d) a + b + c = 2.107. ( )2

cos sin �x �x dxππ − −∫ is equ�l to ( where � �nd q �re integers )(�) π − (b) 0 (c) π (d) 2π.108. If ( ) ( )40cos ,xg x t dt then g x π · +∫ e�uals(a) ( ) ( ) g x g π + (b) ( ) ( ) g x g π − (c) ( ) ( ) g x g π (d) ( ) ( ) / g x g π .M�thiit e� Le�rni ng Res ources doubt@m�thiit.in www.m�thiit.in109. The v�lue of ( )2 101 xe dx−+ ·∫(a) -1 (b) 2 (c) 11 e−+ (d) None of these.110. ( )( ) ( )202� f xdxf x f � x ·+ −∫(�) � (b) 2�

(c) 2�(d) 0.111. The v�lue of 0sinnx dxπ υ +∫

is(a) 2 1 cos n υ + + (b) 2 1 cos n υ + − (c) 2 1 n +(d) 2 cos n υ +.112. If / 420tan ,nn n n� x dx then� �π−· + ·∫(a) 11 n − (b) 11 n + (c) 12 1 n −(d) 12 1 n +.113. 10logsin2 x dxπ | `· . ,∫(a) log 2 − (b) log 2 (c) log 22π(d) log 22π−.114. 12 0log

1xdxx·−∫(a) log 22π (b) log 2 π (c) log 22π−(d) log 2 π − .115. / 20cot x x dxπ∫ eq�als(a) log 22π− (b) log 22π (c) log 2 π (d) log 2 π − .116. The integral val�e ( ) ( ) ( )03 223 3 3 1 cos 1 x x x x x dx− + + + + + +∫ is(a) 2 (b) 4 (c) 0 (d) 8 .117. If ( )12sin11 sin , 0,2 3xt f t dt x x then fπ | ` | `· − ∈ . , . ,∫ eq�al to(a) 3 (b)

13 (c) 13(d) 3118. 201sin sin2nx x dxπ | `− . ,∫ eq�als(a) n (b) 2n (c) � 2n (d) None of these119. The val�e of 31aadxx x−+∫ is(a) 0 (b) 6011adxx +∫ (c) 30121adxx +∫(d) ( )3011a

dxa x + −∫.120. / 3/ 61 tandxxππ ·+∫(a) /12 π (b) / 2 π (c) / 6 π (d) / 4 π121. 44 4sinsin cosxdxx xππ − ·+∫(a) / 4 π (b) / 2 π (c) 3 / 2 π(d) π122. I� � is continuous �unction, then(a) ( ) ( ) ( )2 22 0� x dx � x � x dx− · − − ] ] ∫ ∫(b) ( ) ( )5 103 62 1 f x dx f x dx− −· −∫ ∫(c) ( ) ( )5 4

3 41 f x dx f x dx− −· −∫ ∫(d) ( ) ( )5 63 21 f x dx f x dx− −· −∫ ∫123. The val�e of 2 2 21lim .....1 4 9 2nn n nn n n n→∞ ]+ + + + ]+ + + ] is eq�al to(a) 2π (b) 4π (c) 1 (d) None of these.124. 3 3 3 31 4 1lim .....1 2 2nn n n→∞+ + ++ + is eq�al to(a) 1log 33 e (b) 1log 23 e (c) 1 1log3 3e

(d) None of these.125. 99 99 99 991001 2 3 ....limnnn→∞+ + +·(a) 9100 (b) 1100 (c) 199(d) 1101.126. 1/!limnnnnn→∞ ] ] ] e�uals(a) e (b) 1/ e (c) / 4 π(d) 4/ π.127. 22 211limnnrre�ualsn n r→∞

· +∑(a) 1 5 + (b) 1 5 − + (c) 1 2 − + (d) 1 2 + .Mathiit e� Learni ng Res o�rces do�bt@mathiit.in www.mathiit.inMathiit e� Learni ng Res o�rces do�bt@mathiit.in www.mathiit.in128. 1 1 1 1lim .....1 2 2nn n n n→∞ ]+ + + · ]+ + ](a) 0 (b) log 4e (c) log 3e(d) log 2e.129. 2 21limnnkkn k→∞· +∑ is e�ual to(a) 1log 22 (b) log 2 (c) / 4 π (d) / 2 π .130. ( )2 2 21 1 1 1lim .....2 1nn n n n n n n n→∞ ] ] + + + + ]

+ + + − ] is eq�al to(a) 2 2 2 + (b) 2 2 2 − (c) 2 2 (d) 2.131. 11 2 3 ......lim� � � ��

nnn +→∞+ + + +·(a) 11 � + (b) 11 � − (c) 1 11 � �−−(d) 12 � +.132. 11limrnnnren→∞·∑ is e�ual to(a) 1 e + (b) 1 e − (c) 1� e (d) e.133. The correct eval�ation of 40sin x dxπ∫ is(a) 83π (b) 2

3π (c) 43π(d) 38π.134. The �oints of intersection of ( ) ( )122 5xF x t dt · −∫ and ( )202 ,xF x tdt ·∫ are(a) 6 36,5 25| ` . , (b) 2 4,3 9| ` . , (c) 1 1,3 9| ` . ,(d) 1 1,5 25| ` . ,.135. ( )0"b c� x a dx

−+ ·∫(a) ( ) ( ) � � � a � b − (b) ( ) ( ) ' ' f b c a f a − + − (c) ( ) ( ) ' ' f b c a f a + − + (d) None of these.136. The greatest val�e of the f�nction ( )1xF x t dt · ·∫ on the interval 1 1,2 2 ]− ] ] is given by(a) 38 (b) 12− (c) 38−(d) 25.137. ( )/ 22 2/ 2sin cos sin cos x x x x dxππ − + ·∫(a) 215 (b) 415 (c) 615(d) 815.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in138. ( )

3021dxx x∞·+ +∫(a) 38 (b) 18 (c) 38−(d) None of these.139. If ( ) 2221,xtxf x e dt+−·∫ then ( ) � x increases in(a) ( ) 2, 2 (b) No value o� x (c) ( ) 0, ∞ (d) ( ) , 0 −∞40. If ( ) ( )log,xf x dx xe f x−· +∫ then ( ) � x is(a) 1 (b) 0 (c) xce (d) log x .141. The value o� ( )3/ 20sin cos dπθ θ θ∫ is

(a) 2/9 (b) 2/15 (c) 8/45 (d) 5/2.142. 201log1dxxx x∞| `+ +. ,∫ is e�ual to(a) log 2 π (b) log 2 π − (c) ( ) / 2 log2 π (d) ( ) / 2 log 2 π − .143. ( )20 21xln x dxx∞+∫ is eq�al to(a) 0 (b) 1 (c) ∞ (d) None of these.144. If ( ) 2,1ttdxf tx−·+∫ then ( ) � 1 � is(a) Zero (b) 2 / 3 (c) -1 (d) 1.145. I� ( ) ( )32 log , 0 ,xxF x t dt x · >∫ then ( ) F x ′ ·(a) ( )29 4 log x x x − (b) ( )2

4 9 log x x x − (c) ( )29 4 log x x x + (d) None of these.146. 11202sin1d xdxdx x− ] | ` ]+. , ]∫ is eq�al to(a) 0 (b) π (c) / 2 π(d) / 4 π.147. Let ( ) 212 .xf x t dt · −∫ Then real roots of the eq�ation ( )2' 0 x f x − · are(a) 1 t (b) 12t (c) 12t (d) 0 and 1.Mathiit e- Learni ng Res ources doubt@mathiit.in www.mathiit.in148. ( )( )201 1xdxx x∞·+ +∫(a) 0 (b) / 2 π (c) / 4 π(d) 1.

149. Let ( ) ( ) ( )3sin4sin13; 0. 1 ,xxd eF x x I� e dx F k Fdx x x| `· > · − . , ∫ then one of the �ossible val�e of ‘k’, is(a) 15 (b) 16 (c) 63 (d) 64.150. If ( ) ( )0sin , �xf x t t dt then f x · ·∫(a) cos sin x x x + (b) sin x x (c) cos x x (d) None o� these.151. 2 2 22 2 2 21 1 2 4 1lim sec sec .... sec 1nn n n n n→∞ ]+ + + ] ] e�uals(a) tan1 (b) 1tan12 (c) 1sec12(d) 1cos 12 ec

.152. Area bounded by the curve log , y x · x axis − and the ordinates 1, 2 x x · · is(a) log 4 . s� unit (b) ( ) log 4 1 . s� unit + (c) ( ) log 4 1 . s� unit − (d) None of these.153. Area bo�nded by the �arabola 24 , y x y axis · − and the lines 1, 4 y y · · is(a) 3 . s� unit (b) 7.5 s� unit (c) 7.3 s� unit(d) None o� these.154. I� the ordinate x a · divides the area bounded by the curve 281 , y x axisx| `· + − . , and the ordinates 2, 4 x x · · into two e�ual �arts, then ‘a’ =(a) 8 (b) 2 2 (c) 2 (d) 2 .155. Area bounded by y x s in x · and x axis − between 0 x · and 2 x π · , is(a) 0 (b) 2 . s� unit π (c) . s� unit π (d) 4 . s� unit π .156. Area under the curve 2 cos 2 y s in x x · + between 0 x · and ,4x π· is(a) 2 . s� unit (b) 1 . s� unit (c) 3 . s� unit (d) 4 . s� unit .157. Area under the curve 3 4 y x · + between 0 x · and 4, x · is(a) 56.9 s� unit (b) 64.9 s� unit (c) 8 . s� unit (d) None o� these.158. Area bounded by �arabola 2y x · and straight line 2y x · is(a) 43

(b) 1 (c) 23(d) 13.159. Area bounded by lines 2 , 2 y x y x · + · − and 2 x · is(a) 3 (b) 4 (c) 8 (d)16.160. The ratio o� the areas bounded by the curves cos y x · and cos 2 y x · between 0, / 3 x x π · ·and , x axis − is(a) 2:1 (b) 1: 1 (c) 1 : 2 (d) 2 : 1.161. The area bo�nded by the c�rve 3, y x x axis · − and two ordinates 1 x · to 2 x · e�ual to(a) 15.2 s� unit (b) 15.4 s� unit (c) 17.2 s� unit(d) 17.4 s� unit.162. The area bounded by the x axis − and the c�rve sin y x · and 0 x · , x π · is(a) 1 (b) 2 (c) 3 (d) 4.163. For 0 , x π ≤ ≤ the area bounded by y x · and sin , y x x · + is(a) 2 (b) 4 (c) 2π(d) 4π.164. The area o� the region bounded by the x axis − and the c�rves defined by ( ) tan , / 3 / 3 y x x π · − ≤ ≤ is(a) log 2 (b) log 2 − (c) 2log 2 (d) 0.165. If the area above the x axis − , bo�nded by the c�rves 2kxy · and 0 x · and 2 x · is 3

,2 Inthen the value o� ‘ k ’ is(a) 12 (b) 1 (c) �1 (d) 2.166. The area bounded by the x�axis, the curve ( ) y f x · and the lines 1, x x b · · is e�ual to 21 2 b + − forall 1, b > then ( ) f x is(a) 1 x − (b) 1 x + (c) 21 x +(d) 21xx +.Mathiit e� Learni ng Res o�rces do�bt@mathiit.in www.mathiit.in167. The area bo�nded by the c�rve ( ) y f x · , x axis − and ordinates 1 x · and x b · is ( ) ( ) 1 sin 3 4 , b b − +then ( ) f x is(a) ( ) ( ) ( ) 3 1 cos 3 4 sin 3 4 x x x − + + + (b) ( ) ( ) ( ) 1 sin 3 4 3cos 3 4 b x x − + + + (c) ( ) ( ) ( ) 1 cos 3 4 3sin 3 4 b x x − + + + (d) None of these.168. The area of the region (in the sq�are �nit) bo�nded by the c�rve 24 , x y · line 2 x · and x axis is(a) 1 (b) 23 (c) 43(d) 83.169. Area under the curve 24 y x x · − within the x axis − and the line 2 x · , is(a) 16.3 s� unit (b) 16.

3 s� unit − (c) 4.7 sq �nit(d) Cannot be calc�lated.170. Area bo�nded by the c�rve 3 2 10 0, xy x y x axis − − − · − and the lines 3, 4 x x · · is(a) 16log 2 13 − (b) 16 log 2 3 − (c) 16log 2 3 + (d) None of these.171. The area bo�nded by c�rve 2, 4 y x line y · · and y axis − is(a) 163 (b) 643 (c) 7 2(d) None of these.172. The area bo�nded by the straight lines 0, 2 x x · · and the curves 22 , 2xy y x x · · − is(a) 4 13 log 2− (b) 3 4log 2 3+ (c) 41log 2 −(d)3 4log 2 3− .173. The area between the c�rve 2sin , y x x axis · − and the ordinates 0 x · and 2x π· is(a) 2π (b) 4π (c) 8

π(d) π .174. The area bounded by the circle 2 24, x y + · line 3 x y · and x axis − lying in the first q�adrant, is(a) 2π (b) 4π (c) 3π(d) π .175. The area bo�nded by the c�rve 24 y x x · − and the x axis − , is(a) 30.7 sq �nit (b) 31.7 sq �nit (c) 32.3 sq �nit(d) 34.3 sq �nit.Mathiit e� Learni ng Res o�rces do�bt@mathiit.in www.mathiit.in176. Area of the region bo�nded by the c�rve tan , y x · tangent drawn to the curve at 4x π· and the x axis − is(a) 14 (b) 1log 24+ (c) 1log 24− (d) None of these.

177. The area between the c�rve 24 3 y x x · + − and x axis − is(a) 125 / 6 (b) 125 / 3 (c) 125 / 2 (d) None of these.178. Area inside the �arabola 24 y ax · , between the lines x a · and 4 x a · is e�ual to(a) 24a (b) 28a (c) 2283 a(d) 2353 a.179. The area o� the region bounded by 1 y x · − and 1 y · is(a) 2 (b) 1 (c) 12(d) None o� these.180. The area between the curve 24 , y ax x axis · − and the ordinates 0 x · and x a · is(a) 243 a (b) 283a (c) 223 a(d) 253 a.181. The area o� the curve ( )2 2xy a a x · − bo�nded by y axis − is(a) 2a π (b)

22 a π (c) 23 a π(d) 24 a π.182.The area enclosed by the �arabolas 21 y x · − and 21 y x · − is(a) 1 / 3 (b) 2 / 3 (c) 4 / 3 (d) 8 / 3.183. The area of the smaller segment c�t off from the circle 2 29 x y + · by 1 x · is(a) ( )119sec 3 82−− (b) ( )19sec 3 8−−(c) ( )18 9sec 3−− (d) None of these.184.The area of the region bo�nded by the c�rves 2 , 1, 3 y x x x · − · · and the x axis − is(a) 4 (b) 2 (c) 3 (d) 1.185. The area bo�nded by the c�rves logey x · and ( )2logey x · is (a) 3 e − (b) 3 e − (c) ( )132 e −(d) ( )132 e −.Mathiit e� Learni ng Res o�rcesMathiit e� Learni ng Res o�rces do�bt@mathiit.in www.mathiit.in186. The area of fig�re bo�nded by ,

x xy e y e−· · and the straight line 1 x · is(a) 1ee+ (b) 1ee− (c) 12 ee+ − (d) 12 ee+ + .187. The area bo�nded by the c�rves , 2 3 y x y x · + ·and x axis − in the 1st q�adrant is(a) 9 (b) 274 (c) 36 (d) 18 .188. The area enclosed between the c�rve ( ) logey x e · + and the co - ordinate axes is(a) 3 (b) 4 (c) 1 (d) 2 .189. The �arabolas 24 y x · and 24 x y · divide the s�uare region bounded by the lines 4 x · , 4 y · and thecoordinate axes. I� 1 2 3, , S S S are res�ectively the areas o� these o� these �arts numbered �rom to� to bottom, then 1 2 3: : S S S is(a) 2 : 1 : 2 (b) 1 : 1 : 1 (c) 1: 2 : 1 (d) 1 : 2 : 3.190. I� A is the area o� the region bounded by the curve 3 4, y x x axis · + and the line 1 x · −and B is that areabo�nded by c�rve 23 4, y x x axis · + · and the lines 1 x · − and 4 x · then A : B is e�ual to(a) 1 : 1 (b) 2 : 1 (c) 1: 2 (d) None o� these.191. The area bounded by the curve ( ) ( )2 21 , 1 y x y x · + · − and then line 1

4y · is(a) 1/6 (b) 2/3 (c) 1/4 (d) 1/3.192. Let ( ) � x be a non - negative continous �unction such that the area bounded by the curve ( ) y � x · ,x axis − and the ordinates ,4 4x xπ πβ · · > is sin cos 24πβ β β β| `+ + . , then 2f π | ` . , is(a) 1 24π | `− − . , () 1 24π | `− + . , (c) 2 14π | `+ − . ,(d) 2 14π | `− + . ,.193. Let y e the function which �asses through ( 1, 2 ) having slo�e ( 2x + 1) . The area ounded etween the curveand x axis · is(a) 6 . s� unit (b) 5/ 6 . s� unit (c) 1/ 6 . s� unit (d) None o� these.w w w. m a t h i i t . i n LEVEL - 3 (Tougher Problems)Mathiit e- Learni ng Res ourcesdoubt@mathiit.in www.mathiit.in1. Let ( ) � x be a �unction satis�ying ( ) � x ′= ( ) � x with ( ) 0 1 � · and ( ) g x be the �unction satis�ying

( ) ( ) 2� x g x x + · . The value o� integral ( ) ( )10� x g x dx∫is e�ual to(a) ( )174 e − () ( )124 e − (c) ( )132 e −(d) None of these.2. Let f e a �ositive function . Let 1 21 1( (1 )) , ( (1 ))k kk kI xf x x dx I f x x dx− −· − · −∫ ∫ when 2 1 0. k − > Then 1 2/ I I is(a) 2 () k (c)1/2 (d) 1.3. 714 01xdxx −∫ is equal to(a) 1 () 13 (c) 23(d) 3π.4. If n is any integer, then ( )2

cos 30cos 2 1xe n xdxπ+ ·∫(a) x (b) 1 (c) 0 (d) None o� these.5. Let a, b, c be non-zero real numbers such that ( )( ) ( )( )1 28 2 8 20 01 cos 1 cos x ax bx c dx x ax bx c dx + + + · + + +∫ ∫.Then the �uadratic e�uation 20 ax bx c + + · has(a) No root in (0, 2) (b) At least one root in (0, 2)(c) A double root in (0, 2) (d) None o� these .6. I� ( )1, 1,x� x t dt x−· ≥ −∫ then(a) f and ' f are continous for 1 0 x + > () f is continous ut f ′ is not continous for 1 0 x + >(c) f and f ′ are not continous at 0 x · (d) � is continous at 0 x · but � ′ is not so.7. Let ( ) ( )0xg x � t dt ·∫ where ( ) [ ]11, 0,12 � t t ≤ ≤ ∈ and ( ) 102� t ≤ ≤ �or ( ]1, 2 , t ∈ then(a) ( )3 122 2g − ≤ < () ( ) 0 2 2 g ≤ < (c) ( )

3 522 2g < ≤ (d) ( ) 2 2 4 g < < .8. The value of 2cos, 0,1 xxdx aaππ − >+∫ is(a) π () aπ (c) 2π(d) 2π.9. If ( ) ( ) ( )( )( )1, 1 ,1xf axf aef x I xg x x dxe −· · −+ ∫ and ( ) ( )( )( )2 1 ,f af aI g x x dx−· −∫ then the value of 21II is

(a) 1 () 3 − (c) 1 −(d) 2.10. Let ( ) ( )1 10 01, , f x dx x f x dx a · ·∫ ∫ and ( )12 20, x � x dx a ·∫ then the value o� ( ) ( )1 20x a � x dx − ·∫(a) 0 (b) 2a (c) 21 a − (d) 22 2 a a − + .11. Given that ( )( )( ) ( )( )( )22 2 2 2 2 202x dxa c c a x a x x cπ∞·+ + + + + +∫ then the value o� ( )( )22 204 9x dxx x∞+ +∫ is(a) 60π (b) 20

π (c) 40π(d) 80π.12. I� ( ) ( )10, 1 ,nml m n t t dt · +∫ then the ex�ression �or ( ) , l m n in terms o� ( ) 1, 1 l m n + − is(a) ( )21, 11 1nnl m nm m− + −+ + () ( ) 1, 11nl m nm + −+ (c) ( )21, 11 1nnl m nm m+ + −+ + (d) ( ) 1, 11ml m nn + −+13. 4 4 4 3 3 35 51 2 3 ..... 1 2 3 ....lim limn nn nn n

→∞ →∞+ + + + + + + +− ·(a) 130 (b) Zero (c) 14(d) 15.14. I� ( )2502, 0,5tx� x dx t t · >∫ then 425�

| `· . ,(a) 25 (b) 52 (c) 25−(d) None of these.15. For which of the following values of ‘m’, the area of the region bounded by the curve 2y x x · − and theline y mx · e�uals 92(a) - 4 (b) - 2 (c) 2 (d) 4.16. Area enclosed between the curve ( )2 32 y a x x − · and line 2 x a · above x axis − is(a) 2a π () 232

a π (c) 22 a π (d) 23 a π .17. What is the area ounded y the curves 2 29 x y + · and 28 y x · is(a) 0 (b) 12 2 9 19sin3 2 3π − | `+ − . ,(c) 16π(d) None of these.18. The area ounded y the curves 1 y x · − and 1 y x · − + is(a) 1 () 2 (c) 2 2 (d) 4.Mathiit e� Learni ng Res ources dout@mathiit.in www.mathiit.in19. If for a real numer [ ] , y y is the greatest integer less than or equal to y , then the value of the integral[ ]3 / 2/ 22sin x dxππ∫ is(a) π − () 0 (c) 2π− (d) 2π.20. If ( ) 1sin , ' 22 2xf x A B fπ | ` | `· + · . , . , and ( )1

02,A� x dxπ·∫ then the constants A and B are res�ectively(a) 2 2andπ π (b) 2 3andπ π (c) 40 andπ(d) 40andπ− .21. / 40tan ,nnI x dxπ·∫ then [ ]2lim n nnI I −→∞ + equals(a) 1/ 2 () 1 (c) ∞(d) 0.22. The area ounded y the curves , , y In x y In x y in x · · · and y in x · is(a) 4 . s� unit (b) 6 . s� unit (c) 10 . s� unit (d) None o� these.23. ( )01sin2,

sinn xdx n Nxπ| `+ . ,∈∫ e�uals(a) nπ (b) ( ) 2 12n π+ (c) π (d) 0.24. I� ( )2 100,xe x dx α − ·∫ then(a) 1 2 α < < (b) 0 α < (c) 0 1 α < <(d) None of these.25. 10sin x dxππ∫ is(�) 20 (b) 8 (c) 10 (d) 18.26. ( )22 1 sin1 cosx xdxxππ −++∫ is(�) 2/ 4 π (b) 2

π (c) 0 (d) / 2 π.27. If 2 2 3 1 1 2 231 2 3 40 0 1 12 , 2 , 2 , 2 ,x x xI dx I x dx I dx I dx · · · ·∫ ∫ ∫ ∫ then(a) 3 4I I · (b) 3 4I I > (c) 2 1I I > (d) 1 2I I > .28. I� ( ) 12 3 , � x � xx| `− · . , then ( )21� x dx∫ is e�ual to(a) 325 In (b) ( )31 25 In−+ (c) 325 In−(d) None of these.Mathiit e� Learni ng Res ources dout@mathiit.in www.mathiit.inLEVEL � 1 (Fundamentals of Definite Integration)ANSWER KEYMathiit e� Learni ng Res ources dout@mathiit.in www.mathiit.in

1. c2. a3. c4. 5. c6. a7. 8. d9. a10. c11. d12. c13. a14. d15. a16. 17. c18. d19. d20. c21. 22. a23. c24. 25. a26. c27. c28. 29. c30. d31. d32. a33. 34. 35. d36. 37. a38. a39. c40. c41. 42. 43. a44. 45. a46. c47. 48. c49. 50. a51. d52. a53. a54. a55. 56. c57. 58. c59. 60. d

61. 62. a63. d64. c65. a66. c67. a68. d69. d70. 71. 72. a73. d74. c75. a76. c77. 78. a79. d80. 81. 82. c83. d84. a85. a86. 87. 88. a89. a90. 91. c92. cLEVEL � 2 (Pro�erties of Definite Integration)ANSWER KEYMathiit e� Learni ng Res ources dout@mathiit.in www.mathiit.in1. 2. c3. d4. 5. 6. d7. a8. c9. d10. c11. d12. c13. d14. a15. a16. d17. 18. c19. 20. 21. c22. d23. 24. d25.

26. c27. 28. 29. d30. a31. a32. 33. 34. a35. c36. c37. d38. a39. 40. a41. 42. a43. d44. d45. c46. 47. c48. d49. d50. c51. a52. 53. 54. a55. c56. 57. c58. a59. 60. c61. a62. 63. a64. 65. c66. 67. a68. 69. d70. d71. c72. a73. 74. c75. a76. c77. a78. a79. d80. 81. c82. 83. a84. c85. c

86. c87. c88. a89. 90. c91. c92. a93. d94. d95. d96. 97. c98. c99. d100. a101. 102. c103. c104. c105. c106. c107. d108. a109. d110. a111. 112. a113. a114. c115. 116. 117. a118. 119. a120. a121. d122. d123. 124. 125. 126. 127. 128. d129. a130. 131. a132. 133. d134. a135. 136. c137. 138. a139. d140. c141. c142. a143. a144. d145. a

146. c147. a148. c149. d150. 151. 152. c153. c154. 155. d156. 157. d158. a159. 160. d161. 162. 163. a164. c165. 166. d167. a168. 169. a170. c171. 172. d173. 174. c175. c176. d177. a178. c179. 180. 181. a182. d183. 184. d185. a186. c187. a188. c189. 190. a191. d192. 193. cLEVEL � 3 (Tougher Prolems)ANSWER KEYMathiit e� Learni ng Res ources dout@mathiit.in www.mathiit.in1. d2. c3. 4. c5. 6. a7. 8. c9. d

10. a11. a12. a13. d14. a22. a23. c24. c25. d26. 27. d28. 15. 16. 17. 18. 19. c20. c21.