Indef Integration

UNIT - 4

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Today’s Mathiit’ians..... Tomorrow’s IITi’ians.....

CONTENTS

* Synopsis

Questions

* Level - 1

* Level - 2

* Level - 3

Answers

* Level - 1

* Level - 2

* Level - 3

e-Learning Resources

w w w . m a t h i i t . i n

Indefinite IntegrationMaterial

SYNOPSIS

Indefinite Integration

Mathiit e -Learn ing [email protected] www.mathiit.in

Definitions And Results1. If f & g are functions of x such that g¢(x) = f(x) then ,

∫ f(x) dx = g(x) + c⇔ d

dx{g(x)+c} = f(x), where c is called the constant of integration.

2. STANDARD RESULTS :

(i) ∫ (ax + b)n dx = ( )( )

ax b

a n

n++

+1

1 + c n ¹ -1 (ii) ∫ dx

ax b+ =

1

a ln (ax + b) + c

(iii) ∫ eax+b dx = 1

a eax+b + c (iv) ∫ apx+q dx =

1

p

a

na

px q+

l (a > 0) + c

(v) ∫ sin (ax + b) dx = -1

a cos (ax + b) + c (vi) ∫ cos (ax + b) dx =

1

a sin (ax + b) + c

(vii) ∫ tan(ax + b) dx = 1

a ln sec (ax + b) + c (viii) ∫ cot(ax + b) dx =

1

a ln sin(ax + b)+ c

(ix) ∫ sec² (ax + b) dx = 1

a tan(ax + b) + c (x) ∫ cosec²(ax + b) dx = − 1

acot(ax + b)+ c

(xi) ∫ sec (ax + b) . tan (ax + b) dx = 1

a sec (ax + b) + c

(xii) ∫ cosec (ax + b) . cot (ax + b) dx = − 1

a cosec (ax + b) + c

(xiii) ∫ secx dx = ln (secx + tanx) + c OR ln tan π4 2

+

x+ c

(xiv) ∫ cosec x dx = ln (cosecx - cotx) + c OR ln tan x

2 + c OR - ln (cosecx + cotx)

(xv) ∫ d x

a x2 2− = sin-1

x

a + c (xvi) ∫ d x

a x2 2+ =

1

a tan-1

x

a + c

(xvii) ∫ d x

x x a2 2− =

1

a sec-1

x

a + c

(xviii) ∫ d x

x a2 2+ = ln [ ]x x a+ +2 2

OR sinh-1 x

a + c

(xix) ∫ d x

x a2 2− = ln [ ]x x a+ −2 2

OR cosh-1 x

a + c

(xx) ∫ d x

a x2 2− =

1

2a ln

a x

a x

+−

+ c (xxi) ∫ d x

x a2 2− =

1

2a ln

x a

x a

−+

+ c

(xxii) ∫ a x2 2− dx = x

2 a x2 2− +

a2

2 sin-1

x

a + c

(xxiii) ∫ x a2 2+ dx = x

2 x a2 2+ +

a2

2 sinh-1

x

a + c

xxiv) ∫ x a2 2− dx = x

2 x a2 2− -

a2

2 cosh-1

x

a + c

(xxv) ∫ eax. sin bx dx = e

a b

ax

2 2+ (a sin bx - b cos bx) + c (xxvi) ∫ eax . cos bx dx = e

a b

ax

2 2+ (a cos bx + b sin bx) + c


3. INTEGRALS OF THE TYPE :

(i) ∫ [ f(x)] n f ¢(x) dx OR ∫ [ ]′f x

f xn

( )

( ) dx put f(x) = t & proceed .

(ii)dx

ax bx c2 + +∫ , dx

ax bx c2 + +∫ , ax bx c2 + +∫ dx

Express ax2 + bx + c in the form of perfect square & then apply the standard results .

(iii)px q

ax bx c

++ +∫ 2 dx ,

px q

ax bx c

+

+ +∫ 2

dx . Express px + q = A (differential co-efficient of denominator) + B .

(iv) ∫ ex [f(x) + f ′ (x)] dx = ex . f(x) + c

(v) ∫ [f(x) + x f ′ (x)] dx = x f(x) + c

(vi) ∫ d x

x xn( )+1 n ∈ N Take xn common & put 1 + x-n = t .

(vii) ∫ ( )dx

x xnn

n21

1+−( ) n ∈ N , take xn common & put 1+x-n = tn

(viii)( )

dx

x xn n n1

1+

∫ / take xn common as x and put 1 + x -n = t .

(ix) ∫ d x

a b x+ sin2 OR ∫ d x

a b x+ cos2 OR ∫ d x

a x b x x c xsin sin cos cos2 2+ +

Multiply Num. & Denom. by sec² x & put tan x = t .

(x) ∫ d x

a b x+ sinOR ∫ d x

a b x+ cos OR ∫ d x

a b x c x+ +sin cos Hint :

Convert sines & cosines into their respective tangents of half the angles , put tan x

2 = t

(xi) ∫ a x b x c

x m x n

.cos .sin

.cos .sin

+ ++ +l

dx . Express Nr ≡ A(Dr) + B d

d x (Dr) + c & proceed .

(xii) ∫ x

x K x

2

4 2

1

1

++ +

dx OR ∫ x

x K x

2

4 2

1

1

−+ +

dx where K is any constant . Divide Nr & Dr by x² & proceed

(xiii)dx

ax b px q( )+ +∫ & ( )dx

ax bx c px q2 + + +∫ ; put px + q = t2 .

(xiv) dx

ax b px qx r( )+ + +∫ 2

, put ax + b = 1

t ;

( )dx

ax bx c px qx r2 2+ + + +∫ , put x =

1

t

(xv)x

x

−−∫

αβ

dx or ( ) ( )x x− −∫ α β ; put x = a cos2 θ + b sin2 θ

x

x

−−∫

αβ

dx or ( ) ( )x x− −∫ α β ; put x = a sec2 θ - b tan2 θ

( ) ( )dx

x x− −∫

α β ; put x - a = t2 or x - b = t2 .

w w w . m a t h i i t . i nLEVEL - 1 (Objective)


1. If f(x) = 2 2

3

sin sinx x

x

−∫ where x ≠ 0 then Limit

x → 0 f ′ (x) has the value ;

(A) 0 (B) 1 (C) 2 (D) not defined

2. If 12

+∫ sinx

dx = A sin x

4 4−

π then value of A is :

(A) 2 2 (B) 2 (C) 1

2(D) 4 2

3. If y = ( )

dx

x1 2 3 2+

∫ / and y = 0 when x = 0, then value of y when x = 1 is :

(A) 2

3(B) 2 (C) 3 2 (D)

1

2

4. If cos

cot tan

4 1x

x x

+−∫ dx = A cos 4x + B where A & B are constants, then :

(A) A = - 1/4 & B may have any value (B) A = - 1/8 & B may have any value

(C) A = - 1/2 & B = - 1/4 (D) none of these

5.dx

x5 4+∫ cos = I tan-1 mx

tan2

+ C then :

(A) I = 2/3 (B) m = 3 (C) I = 1/3 (D) m = 2/3

6. Given (a > 0) , 1

x xalog∫ dx = loge a log

e (log

e x) is true for :

(A) x > 1 (B) x > e (C) all x ∈ R (D) no real x .

7.( )cot−

∫1 e

e

x

x dx is equal to :

(A) 1

2 ln (e2x + 1) -

( )cot−1 e

e

x

x + x + c (B)

1

2 ln (e2x + 1) +

( )cot−1 e

e

x

x + x + c

(C) 1

2 ln (e2x + 1) -

( )cot−1 e

e

x

x - x + c (D)

1

2 ln (e2x + 1) +

( )cot−1 e

e

x

x - x + c

8.tan cot

tan cot

− −

− −−+

∫1 1

1 1

x x

x x dx is equal to :

(A) 4

π x tan-1 x +

2

π ln (1 + x2) - x + c (B)

4

π x tan-1 x -

2

π ln (1 + x2) + x + c

(C) 4

π x tan-1 x + 2

π ln (1 + x2) + x + c (D) 4

π x tan-1 x - 2

π ln (1 + x2) - x + c


9. If ( )x

x x

4

2 2

1

1

+

+∫ dx = A ln |x| +

B

x1 2+ + c , where c is the constant of integration then

(A) A = 1 ; B = - 1 (B) A = - 1 ; B = 1 (C) A = 1 ; B = 1 (D) A = - 1 ; B = - 1

10. ∫ l

l

n x

x n x

| |

| |1 + dx equals :

(A) 2

31 + ln x (ln | x | - 2) + c (B)

2

31 + ln x (ln | x | + 2) + c

(C) 1

31 + ln x (ln | x | - 2) + c (D) 21 + ln x (3 ln | x | - 2) + c

11. Antiderivative of sin

sin

2

21

x

x+ w.r.t. x is :

(A) x - 2

2 arctan ( )2 tanx + c (B) x -

1

2 arctan

tan x

2

+ c

(C) x - 2 arctan ( )2 tanx + c (D) x - 2 arctan tan x

2

+ c

12. ∫ sin x . cos x . cos 2x . cos 4x . cos 8x . cos 16 x dx equals :

(A) sin 16

1024

x + c (B) -

cos 32

1024

x + c (C)

cos 32

1096

x + c (D) -

cos 32

1096

x + c

13. x x

x

2 2

21

++∫cos

cosec2 x dx is equal to :

(A) cot x - cot -1 x + c (B) c - cot x + cot -1 x

(C) - tan -1 x - cos

sec

ec x

x + c (D) - e n xl tan−1 - cot x + c

where 'c' is constant of integration .

14.3 5

4 5

e e

e e

x x

x x

+−

−

−∫ dx = Ax + B ln | 4e2x - 5 | + c then :

(A) A = -1, B = -7/8; C = const. of integration

(B) A = 1, B = 7/8; C = const. of integration

(C) A = -1/8, B = 7/8 ; C = const. of integration

(D) A = -1, B = 7/8 ; C = const. of integration

15.x

x x

−+∫

1

1

12

. dx equals :

(A) sin -1 1

x +

x

x

2 1−(B)

x

x

2 1− + cos -1

1

x + c

(C) sec -1 x - x

x

2 1− + c (D) tan -1 x2 1+ -

x

x

2 1− + c

16.dx

x x−∫ 2

equals :

(A) 2 sin -1 x + c (B) sin -1 (2x - 1) + c (C) c - 2 cos -1 (2x - 1) (D) cos -1 2 x x− 2 + c


17. ∫ 2mx . 3nx dx when m, n ∈ N is equal to :

(A) 2 3

2 3

mx nx

m n n n

++l l

+ c (B) ( )e

m n n n

m n n n xl l

l l

2 3

2 3

+

+ + c

(C) ( )2 3

2 3

mx nx

m nn

.

.l + c (D)

( )mn

m n n n

x x. .2 3

2 3l l+ + c

18.dx

x xcos . sin3 2∫ equals :

(A) 2

5 (tan x)5/2 + 2 tanx + c (B)

2

5 (tan2 x + 5) tanx + c

(C) 2

5 (tan2 x + 5) 2tanx + c (D) none

19. If dx

x xsin cos3 5∫ = a cot x + b tan3 x + c where c is an arbitrary constant of integration then

the values of ‘a’ and ‘b’ are respectively :

(A) - 2 & 2

3(B) 2 & -

2

3 (C) 2 &

2

3 (D) none

20. If ∫ eu . sin 2x dx can be found in terms of known functions of x then u can be :

(A) x (B) sin x (C) cos x (D) cos 2x

21. cos cossin sin

3 5

2 4

x xx x

++∫ dx :

(A) sin x - 6 tan-1 (sin x) + c (B) sin x - 2 sin-1 x + c

(C) sin x - 2 (sin x)-1 - 6 tan-1 (sin x) + c (D) sin x - 2 (sin x)-1 + 5 tan-1 (sin x) + c

22. ln x

x x

(tan )

sin cos∫ dx equal :

(A) 1

2 ln2 (cot x) + c (B)

1

2 ln2 (sec x) + c

(C) 1

2 ln2 (sin x sec x) + c (D)

1

2 ln2 (cos x cosec x) + c

23. ∫ sec2 24

x −

π dx equals :

(A) c - 1

2 cot 2

4x +

π(B)

1

2 tan 2

4x −

π + c

(C) 1

2(tan 4x - sec 4x) + c (D) none

24. Primitive of

( )3 1

1

4

4 2

x

x x

−

+ + w.r.t. x is :

(A) x

x x4 1+ + + c (B) -

x

x x4 1+ + + c (C)

x

x x

++ +

1

14 + c (D) - x

x x

++ +

1

14 + c


1. ∫ cos cos

cos

5 4

1 2 3

x x

xdx

+−

2.∫ cos x . ex. x2 dx 3. ∫ sin( )

sin( )

x a

x a

−+

dx

4. ∫ cot

( sin ) (sec )

x dx

x x1 1− +5.∫ cos cot

cos cot.

sec

sec

ec x x

ec x x

x

x

−+ +1 2

dx 6. ∫d x

x xsin sec+

7.∫ tan x . tan 2x . tan 3x dx 8. ( )dx

x xsin sin 2 +∫

α9. ∫ x

x x x

2

2( sin cos )+ dx

10.∫( )ln x x

x

cos cos

sin

+ 22 dx 11.

sin

sin cos

x

x x+∫ dx 12.∫ ex x x

xxsin .

cos sin

cos

3

2

− dx

13.∫ ( )d x

a b x+ cos2 (a > b) 14. ∫

cos

sin

2 x

x dx 15.

cot tan

sin

x x

x

−+∫ 1 3 2

dx

16. ( )5 4

1

4 5

5 2

x x

x x

+

+ +∫ dx 17.

( )dx

x4 21−

∫ 18.∫ ex ( )x

x

2

2

1

1

++( )

dx

19.∫ x x+ +2 2 dx 20.∫ ( )[ ]x l n x x

x

2 2

4

1 1 2+ + −

ln dx 21.∫ l n x

x(ln )

(ln )+

12 dx

22.∫( ) ( )[ ]

dx

x x− +1 23 5 1 4/ 23. ( )

dx

x x x x x3 2 23 3 1 2 3+ + + + −∫ 24. ∫

( )

( )

ax b dx

x c x ax b

2

2 2 2 2

−

− +

25. ∫( )e x

x x

x 2

1 1

2

2

−

− −( ) dx 26. ∫ ( )

x

x x7 10 2 3 2− −

/ dx 27. ∫ ( )x x

x

ln/2 3 2

1− dx

28.∫ 1

13

−+

x

x

dx

x29.

2 3

2 3

1

1

−+

+−∫

x

x

x

x dx 30. ∫ ( )

x

x x

d x

x

++ + +

2

3 3 12

31.dx

x x3 31( )+∫ 32.∫ 2 2

2

− −x x

x dx 33. ∫ d x

x x x( ) ( ) ( )− − −α α β

34. Integrate 1

2f ′ (x) w.r.t. x4 , where f (x) = tan -1x + ln 1+ x - ln 1− x

w w w . m a t h i i t . i nLEVEL - 2 (Subjective)

w w w . m a t h i i t . i nLEVEL - 3 (Questions asked from previous Engineering Exams)


1. Find the indefinite integral ∫ [ ]1 11 3 1 4

1 6

1 3 1 2( ) ( )

( )

( ) ( )/ /

/

/ /x x

l n x

x x++

++

dx

2. Evaluate ∫ 3 1

1 13

x

x x

+− +( ) . ( )

dx .

3. Evaluate ∫ f x

x

( )3 1−

dx ; where f(x) is a polynomial of second degree in x such that

f (0)

=

f

(1)

=

3 f

(2)

=

- 3 .

4. Evaluate , ∫ cos 2θ . ln cos sin

cos sin

θ θθ θ

+−

dθ .

5. Evaluate ∫ cos . sin

cos . ( cos )

2 4

1 24 2

x x

x x+ dx .

6. Integrate , ( )

x x

x x

3

2 2

3 2

1 1

+ +

+ +∫

( ) dx .

7. Let f (x) = ∫ ex (x - 1) (x - 2) d x then f decreases in the interval :

(A) (- ∞ , 2) (B) (- 2, - 1) (C) (1, 2) (D) (2, ∞ )

8. Evaluate , ∫ sin -1 2 2

4 8 132

x

x x

+

+ +

d

x .

1. b

2. d

3. d

4. b

5. a,b

6. b

7. c

8. d

9. c

10. a

11. a

12. b

LEVEL - 1 (Objective questions)

ANSWER KEY


13. b,c,d

14. d

15. c

16. a,b,d

17. b,c

18. b

19. a

20. c

21. c

22. a,c,d

23. a,b,c

24. b

1. − +(sinsin

)xx2

2+ c

2. 1

2 ex ( )[ ]x x x x2 21 1− + −cos ( ) . sin + c

3. cos a . arc cos cos

cos

x

a

- sin a . ln ( )sin sin sinx x a+ −2 2 + c

4.1

2 ln tan

x

2 +

1

4 sec²

x

2 + tan

x

2 + c

5. sin-1 1

2 22sec

x

+ c

6. 1

2 3

3

3l n

x x

x xarc x x c

+ −− +

+ + +sin cos

sin costan (sin cos )

7. − − +

l l ln x n x n x(sec ) (sec ) (sec )1

22

1

33 + c

8. -1

sinα ln [ ]cot cot cot cot cotx x x+ + + −α α2 2 1 + c

9. sin cos

sin cos

x x x

x x xc

−+

+

10.cos

sin

2x

x - x - cot x . ln ( )( )e x xcos cos+ 2 + c

11. ln (1 + t) - 1

4 ln (1 + t4) +

1

2 2 ln t t

t t

2

2

2 1

2 1

− ++ +

- 1

2 tan-1 t2 + c where t = cot x

12. esinx (x - secx) + c

13. ( ) ( )−

− ++

−

−+

b x

a b a b x

a

a barc

a b

a b

xsin

( cos )tan . tan/2 2 2 2 3 2

2

2 + c

14. - ( )1

2 2

2 1

2 11

2

2

2l ln

x x

x xn x x

cot cot

cot cotcot cot

− −

+ −+ − +

+ c

LEVEL - 2 (Subjective Questions)

ANSWER KEY


15. tan-1 2 2sin

sin cos

x

x x+

+ c 16. - x

x x

++ +

1

15 + c

17. 3

8 tan-1 x - ( )

x

x4 14 − -

3

16 ln

x

x

−+

1

1 + c 18. ex

x

x

−+

1

1 + c

19.1

3 ( )x x+ +2

3 2

2/

- ( )2

221 2

x x+ +/ + c 20.

( )x x

x x

2 2

3 2

1 1

92 3 1

1+ +− +

. ln

21. xln (lnx) - x

l n x + c 22. 4

3

1

2

1 4x

xc

−+

+

/

23.x x

x

2

2

2 3

8 1

+ −+( )

+ 1

16 . cos-1

2

1x +

+ c 24. sin− +

+1

2ax b

cxk

25. ex 1

1

+−

x

x + c 26. 2 7 20

9 7 10 2

( )x

x xc

−

− −+

27 arc xln x

xcsec −

−+

2 1

28. lnu

u u

uc where u

x

x

| |tan

2

4 2

12

31

13

1 2

3

1

1

−

+ ++ + + = −

+−

29. ( )8

3

1

2 5

5 1

5 111 1 2tan sin− −+ −

+

− − −t nt

tx xl + c where t =

1

1

+−

x

x

30. 2

3 3 1arc

x

xctan

( )++

31. 15 5 2

4 1

2

2

x x

x x

+ −+

+ 15

8 ln

1 1

1 1

+ −+ +

x

x + c

32. −− −

+− + − −

− +

−2 2

4

4 2 2 2 2 1

3

2 21x x

xn

x x x

x

xl sin + c

33.−−

−−

+2

α ββα

.x

xc

34. - ln (1 - x4) + c


1. I = I1 + I

2 + c , where ;

Iy y y y y y y

y n y1

8 7 6 5 4 3 2

128

8

7

28

6

56

5

70

4

56

3

28

28 1= − + − + − + − +

; where y = x1/12 + 1

( )I e z e z e z z cz z z2

3 2 222

1

39

1

218 1 3= −

− −

+ − − + ; where z = ln (1 + x1/6)

2. 1

4

1

1

1

2 1 2l n

x

x

x

xc

+−

− +−

+( )

3. lnx x

x

2 1

1

+ +−

+

2

3 arctan2 1

3

x +

+ c

4. (b) 1

2 (sin 2 θ ) ln cos sin

cos sin

θ θθ θ

+−

-

1

2 ln (sec 2 θ ) + c

5. 2 ln (1 + cos 2x) + 2

1 2+ cos x- ln (1 + cos2 2x) + c

6.3

2 tan-1 x -

1

2 ln (1 + x) +

1

4 ln (1 + x2) +

x

x1 2+ + c

7. C

8. -3

2

2 2

3

2 2

3

1

24 8 131 2x x

x x+ +

− + +

−tan log( )

LEVEL - 3 (Questions asked from previous Engineering Exams)

ANSWER KEY


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Indef Integration