INTEGRALS - einstein classeseinsteinclasses.com/Bluetooth Folder/Integeration.pdf · m 15. 9 4x2 x 16. e2x + 3 ... 1 sin2x cosx sinx ... NCERT Solved examples upto the section 7.4

MI – 1

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road

New Delhi – 110 018, Ph. : 9312629035, 8527112111

INTEGRALS

NCERT Solved examples upto the section 7.1 (Introduction) and 7.2 (Integration as an InverseProcess of Differentiation) :

Example 1 : Write an anti derivative for each of the following function using the method ofinspection

(i) cos 2x (ii) 3x2 + 4x3 (iii) 0x,x

1

Solution : (i) x2sin2

1 (ii) x3 + x4

Example 2 : Find the following integrals

(i)

dxx

1x2

3

(ii) dx)1x( 3

2

(iii) dx)x

1e2x( x2

3

Solution : (i) Cx

1

2

x2

(ii) Cxx5

3 3

5

(iii) C|x|loge2x5

2 x2

5

Example 3 : Find the following integrals

(i) dx)xcosx(sin (ii) dx)xcotecx(cosecxcos

(iii)

dxxcos

xsin12

Solution : (i) –cos x + sin x + C (ii) – cot x – cosec x + C (iii) tan x – sec x + C

Example 4 : Find the anti derivative F of f defined by f(x) = 4x3 – 6, where F(0) = 3.

Solution : F(x) = x4 – 6x + 3

EXERCISE 7.1

Find an anti derivative (or integral) of the following functions by the method of inspection

1. sin 2x 2. cos 3x 3. e2x 4. (ax + b)2

5. sin 2x – 4 e3x

Find the following integrals :

6. dx)1e4( x3 7.

dx

x

11x

22

8. dx)cbxax( 2

9. dx)ex2( x210.

dx

x

1x

2

11.

dxx

4x5x2

23

12.

dxx

4x3x3

13.

dx

1x

1xxx 23

14. dxx)x1(

15. dx)3x2x3(x 216. dx)excos3x2( x

17. dx)x5xsin3x2( 2

18. dx)xtanx(secxsec 19. dxxeccos

xsec2

2

20.

dxxcos

xsin322

MI – 2


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Choose the correct answer in questions 21 and 22.

21. The anti derivative of

x

1x equals

(a) Cx2x3

12

1

3

1

(b) Cx2

1x

3

2 23

2

(c) Cx2x3

22

1

2

3

(d) Cx2

1x

2

32

1

2

3

22. If 4

3

x

3x4)x(f

dx

d such that f(2) = 0. Then f(x) is

(a)8

129

x

1x

3

4 (b)8

129

x

1x

4

3

(c)8

129

x

1x

3

4 (d)8

129

x

1x

4

3

Answers :

1. x2cos2

1 2. x3sin

3

13.

x2e2

1

4. 3)bax(

a3

1 5.

x3e3

4x2cos

2

1 6. Cxe

3

4 x3

7. Cx3

x3

8. Ccx2

bx

3

ax 23

9. Cex3

2 x3

10. Cx2|x|log2

x2

11. Cx

4x5

2

x2

12. Cx8x2x7

22

3

2

7

13. Cx3

x3

14. Cx5

2x

3

22

5

2

3

15. Cx2x5

4x

7

62

3

2

5

2

7

16. x2 – 3sinx + ex + C

17. Cx3

10xcos3x

3

22

33 18. tan x + sec x + C

19. tan x – x + C 20. 2 tan x – 3 sec x + C 21. c

22. a

NCERT Solved examples upto the section 7.3 (Methods of Integration)

Example 5 : Integrate the following functions w.r.t. x :

(i) sin mx (ii) 2x sin (x2 + 1) (iii)x

xsecxtan 24

(iv) 2

1

x1

)xsin(tan

MI – 3


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Solution : (i) Cmxcosm

1 (ii) – cos (x2 + 1) + C (iii) Cxtan

5

2 5 (iv) – cos (tan–1x) + C

Example 6 : Find the following integrals :

(i) dxxcosxsin 23 (ii)

dx)axsin(

xsin (iii)

dxxtan1

1

Solution : (i) Cxcos5

1xcos

3

1 53 (ii) x cos a – sin a log |sin (x + a)| + C

(iii) C|xsinxcos|log2

1

2

x ]

EXERCISE 7.2

Integrate the following functions :

1. 2x1

x2

2.

x

)x(log 2

3.xlogxx

1

4. sin x sin (cos x) 5. sin (ax + b) cos (ax + b) 6. bax

7. 2xx 8. 2x21x 9. 1xx)2x4( 2

10.xx

1

11. 0x,

4x

x

12. (x3 – 1)1/3x5

13.33

2

)x32(

x

14. 0x,

)x(logx

1m

15.2x49

x

16. e2x + 3 17.2xe

x18.

2

xtan

x1

e1

19.1e

1ex2

x2

20. x2x2

x2x2

ee

ee

21. tan2(2x – 3)

22. sec2 (7 – 4x) 23.2

1

x1

xsin

24.xsin4xcos6

xsin3xcos2

25. 22 )xtan1(xcos

1

26.

x

xcos27. x2cosx2sin

28.xsin1

xcos

29. cot x log sin x 30.

xcos1

xsin

31.2)xcos1(

xsin

32.

xcot1

1

33.

xtan1

1

34.xcosxsin

xtan35.

x

)xlog1( 236.

x

)xlogx)(1x( 2

37.8

413

x1

)xsin(tanx

MI – 4


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Choose the correct answer in questions 38 and 39.

38.

x10e

x9

10x

dxlog10x10 10

equals

(a) 10x – x10 + C (b) 10x + x10 + C

(c) (10x – x10)–1 + C (d) log(10x + x10) + C

39. xcosxsin

dx22

equals

(a) tan x + cot x + C (b) tan x – cot x + C

(c) tan x cot x + C (d) tan x – cot 2x + C

Answers :

1. log (1 + x2) + C 2. C|)x|(log3

1 3 3. log|1 + logx| + C

4. cos (cos x) + C 5. C)bax(2cosa4

1 6. C)bax(

a3

22

3

7. C)2x(3

4)2x(

5

22

3

2

5

8. C)x21(6

12

32 9. C)1xx(

3

42

32

10. C|1x|log2 11. C)8x(4x3

2 12. C)1x(

4

1)1x(

7

1 3

433

73

13. C)x32(18

123

14. C

m1

)x(log m1

15. |x49|log8

1 2

16. Ce2

1 3x2 17. Ce2

12x

18. Ce xtan 1

19. log (ex + e–x) + C

20. C)eelog(2

1 x2x2 21. Cx)3x2tan(2

1 22. C)x47tan(

4

1

23. C)x(sin2

1 21 24. C|xcos3xsin2|log2

1

25. C)xtan1(

1

26. Cxsin2 27. C)x2(sin

3

12

3

28. Cxsin12 29. C)xsin(log2

1 2 30. –log |1 + cos x|

31. Cxcos1

1

32. C|xsinxcos|log

2

1

2

x

33. C|xsinxcos|log2

1

2

x 34. Cxtan2 35. C)xlog1(

3

1 3

36. C)xlogx(3

1 3 37. C)xcos(tan4

1 41 ] 38. d

39. b

MI – 5


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Example 7 : Find (i) dxxcos2 (ii) dxx3cosx2sin (iii) dxxsin3.

Solution : (i) Cx2sin4

1

2

x (ii) Cxcos

2

1x5cos

10

1 (iii) Cxcos

3

1xcos 3

EXERCISE 7.3

Find the integrals of the following functions :

1. sin2 (2x + 5) 2. sin 3x cos 4x 3. cos 2x cos 4x cos 6x

4. sin3 (2x + 1) 5. sin3x cos3x 6. sin x sin 2x sin 3x

7. sin 4x sin 8x 8.xcos1

xcos1

9.

xcos1

xcos

10. sin4 x 11. cos4 2x 12.xcos1

xsin2

13.

cosxcos

2cosx2cos14.

x2sin1

xsinxcos

15. tan3 2x sec 2x

16. tan4x 17.xcosxsin

xcosxsin22

33 18.

xcos

xsin2x2cos2

2

19.xcosxsin

13

20.2)xsinx(cos

x2cos

21. sin–1(cos x)

22.)bxcos()axcos(

1

Choose the correct answer in questions 23 and 24

23.

dxxcosxsin

xcosxsin22

22

is equal to

(a) tan x + cot x + C (b) tan x + cosec x + C

(c) –tan x + cot x + C (d) tan x + sec x + C

24.

dx)xe(cos

)x1(ex2

x

equals

(a) –cot (exx) + C (b) tan (xex) + C

(c) tan (ex) + C (d) cot (ex) + C

Answers :

1. C)10x4sin(8

1

2

x 2. Cxcos

2

1x7cos

14

1

3. Cx4sin4

1x8sin

8

1xx12sin

12

1

4

1

4. C)1x2(cos

6

1)1x2cos(

2

1 3

5. Cxcos4

1xcos

6

1 46 6. Cx2cos2

1x4cos

4

1x6cos

6

1

4

1

7. Cx12sin12

1x4sin

4

1

2

1

8. Cx

2

xtan2

MI – 6


New Delhi – 110 018, Ph. : 9312629035, 8527112111

9. C2

xtanx 10. Cx4sin

32

1x2sin

4

1

8

x3

11. Cx8sin64

1x4sin

8

1

8

x3 12. x – sin x + C

13. 2(sin x + x cos ) + C 14. Cxsinxcos

1

15. Cx2sec2

1x2sec

6

1 3 16. Cxxtanxtan3

1 3

17. sec x – cosec x + C 18. tan x + C

19. Cxtan2

1|xtan|log 2 20. log |cos x + sin x| + C

21. C2

x

2

x 2

22. C)bxcos(

)axcos(log

)basin(

1

23. a 24. b

NCERT Solved examples upto the section 7.4 (Integrals of Some Particular Functions) :


(i) 16x

dx2

(ii) 2xx2

dx

Solution : (i) C4x

4xlog

8

1

(ii) sin–1(x – 1) + C


(i) 13x6x

dx2 (ii) 10x13x3

dx2

(iii) x2x5

dx

2

Solutkon : (i) C2

3xtan

2

1 1 (ii) C

5x

2x3log

17

1

(iii) C

5

x2x

5

1xlog

5

1 2


(i)

dx

5x6x2

2x(ii)

dx

xx45

3x

2

Solution : (i) C)3x2(tan2

1|5x6x2|log

4

1 12 (ii) C3

2xsinxx45 12

EXERCISE 7.4

Integrate the functions 1 to 23.

1.1x

x36

2

2.

2x41

1

3.

1)x2(

1

2

4.2x259

1

5. 4x21

x3

6. 6

2

x1

x

MI – 7


New Delhi – 110 018, Ph. : 9312629035, 8527112111

7.

1x

1x

2

8.

66

2

ax

x

9.

4xtan

xsec

2

2

10.2x2x

1

2 11.

5x6x9

12

12.2xx67

1

13.)2x)(1x(

1

14.

2xx38

1

15.

)bx)(ax(

1

16.3xx2

1x4

2

17.

1x

2x

2

18. 2x3x21

2x5

19.)4x)(5x(

7x6

20.

2xx4

2x

21.

3x2x

2x

2

22.5x2x

3x2

23.

10x4x

3x5

2

Choose the correct answer in Exercises 24 and 25.

24 2x2x

dx2

equals

(a) x tan–1 (x + 1) + C (b) tan–1 (x + 1) + C

(c) (x + 1) tan–1x + C (d) tan–1x + C

25. 2x4x9

dx equals

(a) C8

8x9sin

9

1 1

(b) C

9

9x8sin

2

1 1

(c) C8

8x9sin

3

1 1

(d) C9

8x9sin

2

1 1

Answers :

1. tan–1x3 + C 2. Cx41x2log2

1 2 3. C5x4xx2

1log

2

4. C3

x5sin

5

1 1 5. Cx2tan

22

3 21 6. C

x1

x1log

6

13

3

7. C1xxlog1x 22 8. Caxxlog3

1 663

9. C4xtanxtanlog 2 10. C2x2x1xlog 2

11. C2

1x3tan

6

1 1

12. C2

3xsin 1

13. C2x3x2

3xlog 2

MI – 8


New Delhi – 110 018, Ph. : 9312629035, 8527112111

14. C14

3x2sin 1

15. C)bx)(ax(

2

baxlog

16. C3xx22 2 17. C1xxlog21x 22

18. C2

1x3tan

23

111x2x3log

6

5 12

19. C20x9x2

9xlog3420x9x6 22

20. C2

2xsin4xx4 12

21. C3x2x1xlog3x2x 22

22. C61x

61xlog

6

25x2xlog

2

1 2

23. C10x4x2xlog710x4x5 22 ]

24. b 25. b

NCERT Solved examples upto the section 7.5 (Integration by Partial Fractions) :

Example 11 : Find )2x)(1x(

dx.

Solution : [ C2x

1xlog

Example 12 : Find

6x5x

1x2

2

dx.

Solution : x – 5 log |x – 2| + 10 log |x – 3| + C

Example 13 : Find

dx

3x()1x(

2x32

.

Solution : C)1x(2

5

3x

1xlog

4

11

Example 14 : Find dx

)4x)(1x(

x22

2

Solution : C2

xtan

3

2xtan

3

1 11

MI – 9


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Example 15 : Find

d

sin4cos5

cos)2sin3(2

.

Solution : Csin2

4)sin2log(3

Example 16 : Find

)1x)(2x(

1xx2

2

.

Solution : Cxtan5

1|1x|log

5

1|2x|log

5

3 12

EXERCISE 7.5

Integrate the rational function in Exercises 1 to 21.

1.)2x)(1x(

x

2.

9x

12

3.)3x)(2x)(1x(

1x3

4.)3x)(2x)(1x(

x

5.

2x3x

x22

6.)x21(x

x1 2

7.)1x)(1x(

x2

8.)2x()1x(

x2

9.1xxx

5x323

10.)3x2)(1x(

3x22

11.

)4x)(1x(

x52

12.1x

1xx2

3

13.)x1)(x1(

22

14.2)2x(

1x3

15.

1x

14

16.)1x(x

1n

17.)xsin2)(xsin1(

xcos

18.)4x)(3x(

)2x)(1x(22

22

19.

)3x)(1x(

x222

20.)1x(x

14

21.)1e(

1x

Choose the correct answer in each of the Exercises 22 and 23.

22. equals

)2x)(1x(

xdx

(a) C2x

)1x(log

2

(b) C

1x

)2x(log

2

(c) C2x

1xlog

2

(d) log |(x – 1) (x – 2)| + C

MI – 10


New Delhi – 110 018, Ph. : 9312629035, 8527112111

23. equals

)1x(x

dx2

(a) C)1xlog(2

1|x|log 2 (b) C)1xlog(

2

1|x|log 2

(c) C)1xlog(2

1|x|log 2 (d) C)1xlog(|x|log 2

Answers :

1. C|1x|

)2x(log

2

2. C

3x

3xlog

6

1

3. log |x – 1| – 5 log |x – 2| + 4 log |x – 3| + C

4. C|3x|log2

3|2x|log21xlog

2

1 5. 4log |x + 2| – 2 log |x + 1| + C

6. C|x21|log4

3xlog

2

x 7. Cxtan

2

1)1xlog(

4

11xlog

2

1 12

8. C)1x(3

1

2x

1xlog

9

2

9. C

1x

4

1x

1xlog

9

1

10. C|3x2|log5

12|1–x|log

10

11xlog

2

5

11. C|2x|log6

5|2x|log

2

51xlog

3

5

12. C|1x|log2

3|1x|log

2

1

2

x2

13. Cxtan)x1log(2

1|1x|log 12

14. C2x

7|2x|log3

15. Cxtan

2

1

1x

1xlog

4

1 1

16. C1x

xlog

n

1n

n

17. Cxsin1

xsin2log

18. C2

xtan3

3

xtan

3

2x 11

19. C3x

1xlog

2

12

2

20. Cx

1xlog

4

14

4

21. Ce

1elog

x

x

22. b 23. a

MI – 11


New Delhi – 110 018, Ph. : 9312629035, 8527112111

NCERT Solved examples upto the section 7.6 (Integration by Parts) :

Example 17 : Find dxxcosx

Solution : dx2

xxsin

2

x)x(cos

22

Example 18 : Find dxxlog .

Solution : x log x – x + C

Example 19 : Find dxxex.

Solution : xex – ex + C

Example 20 : Find

dxx1

xsinx

2

1

.

Solution : Cxsinx1x 12

Example 21 : Find dxxsinex.

Solution : C)xcosx(sin2

ex

Example 22 : Find (i) dxx1

1xtane

2

1x

(ii)

dx

)1x(

e)1x(2

x2

.

Solution : (i) ex tan–1x + C (ii) Ce1x

1x x

EXERCISE 7.6

Integrate the functions in Exercises 1 to 22.

1. x sin x 2. x sin 3x 3. x2 ex 4. x log x

5. x log 2x 6. x2 log x 7. x sin–1x 8. x tan–1x

9. x cos–1 x 10. (sin–1x)2 11.2

1

x1

xcosx

12. x sec2x

13. tan–1x 14. x (log x)2 15. (x2 + 1) log x

16. ex (sinx + cosx) 17.2

x

)x1(

xe

18.

xcos1

xsin1ex 19.

2x

x

1

x

1e

20.3

x

)1x(

e)3x(

21. e2x sin x 22.

2

1

x1

x2sin

Choose the correct answer in Exercises 23 and 24

23. dxex3x2 equals

(a) Ce3

1 3x (b) Ce3

1 2x (c) Ce2

1 3x (d) Ce2

1 2x

MI – 12


New Delhi – 110 018, Ph. : 9312629035, 8527112111

24. dx)xtan1(xsecex equals

(a) ex cos x + C (b) ex sec x + C (c) ex sin x + C (d) ex tan x + C

Answers :

1. – x cos x + sin x + C 2. Cx3sin9

1x3cos

3

x

3. ex (x2 – 2x + 2) + C 4. C4

xxlog

2

x 22

5. C4

xx2log

2

x 22

6. C9

xxlog

3

x 33

7. C4

x1xxsin)1x2(

4

1 212

8. Cxtan

2

1

2

xxtan

2

x 112

9. Cx14

x

4

xcos)1x2( 2

12

10. Cx2xsinx12x)x(sin 1221

11. Cxxcosx1 12

12. x tan x + log |cos x| + C

13. C)x1log(2

1xtanx 21 14. C

4

xxlog

2

x)x(log

2

x 222

2

15. Cx9

xxlogx

3

x 33

16. ex sin x + C

17. Cx1

ex

18. C2

xtanex

19. Cx

ex

20. C)1x(

e2

x

21. C)xcosxsin2(5

e x2

22. 2x tan–1x – log (1 + x2) + C

23. a 24. b

Example 23 : Find dx5x2x2 .

Solution : C5x2x1xlog25x2x)1x(2

1 22

Example 24 : Find dxxx23 2.

Solution : C2

1xsin2xx23)1x(

2

1 12

MI – 13


New Delhi – 110 018, Ph. : 9312629035, 8527112111

EXERCISE 7.7


1. 2x4 2. 2x41 3. 6x4x2 4. 1x4x2

5. 2xx41 6. 5x4x2 7. 2xx31 8. x3x2

9.9

x1

2

Choose the correct answer in Exercises 10 to 11.

10. dxx1 2 is equal to

(a) Cx1xlog2

1x1

2

x 22

(b) C)x1(

3

22

32

(c) C)x1(x3

22

32 (d) Cx1xlogx

2

1x1

2

x 2222

11. dx7x8x2 is equal to

(a) C7x8x4xlog97x8x)4x(2

1 22

(b) C7x8x4xlog97x8x)4x(2

1 22

(c) C7x8x4xlog237x8x)4x(2

1 22

(d) C7x8x4xlog2

97x8x)4x(

2

1 22

Answers :

1. C2

xsin2x4x

2

1 12

2. Cx41x2

1x2sin

4

1 21

3. C6x4x2xlog6x4x2

)2x( 22

4. C1x4x2xlog2

31x4x

2

)2x( 22

5. Cxx412

2x

5

2xsin

2

5 21

6. C5x4x2xlog2

95x4x

2

)2x( 22

MI – 14


New Delhi – 110 018, Ph. : 9312629035, 8527112111

7. C13

3x2sin

8

13xx31

4

)3x2( 12

8. Cx3x2

3xlog

8

9x3x

4

3x2 22

9. C9xxlog2

39x

6

x 22

10. a

11. d

NCERT Solved examples upto the section 7.7 (Definite Integral) :

Example 25 : Find

2

0

2 dx)1x( as the limit of a sum.

Solution : 3

14

Example 26 : Evaluate dxe

2

0

x

as the limit of a sum.

Solution : e2 – 1

EXERCISE 7.8

Evaluate the following definite integrals as limit of sums.

1. b

a

dxx 2.

5

0

dx)1x( 3. 3

2

2 dxx 4.

4

1

2 dx)xx(

5.

1

1

2 dxe 6.

4

0

x2 dx)ex(

Answers :

1. )ab(2

1 22 2. 2

353.

3

194.

2

27

5.e

1e 6.

2

e15 8

NCERT Solved examples upto the section 7.8 (Fundamental Theorem of Calculus) :

Example 27 : Evaluate the following :

(i) 3

2

2 dxx (ii)

9

4 22

3dx

)x30(

x(iii)

2

1)2x)(1x(

xdx

(iv)

4

0

3 dtt2cost2sin

MI – 15


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Solution : (i) 3

19 (ii)

99

19 (iii)

27

32log (iv)

8

1

EXERCISE 7.9

Evaluate the definite integrals.

1.

1

1

dx)1x( 2. 3

2

dxx

13.

2

1

23 dx)9x6x5x4(

4.

4

0

dxx2sin 5.

2

0

dxx2cos 6. 5

4

x dxe

7.

4

0

dxxtan 8.

4

6

dxxeccos 9.

1

02x1

dx

10.

1

0

2x1

dx11.

3

2

2 1x

dx12.

2

0

2 dxxcos

13.

3

2

2 1x

xdx14.

1

0

2dx

1x5

3x215.

1

0

x dxxe2

16.

2

1

2

2

3x4x

x517.

4

0

32 dx)2xxsec2( 18.

0

22 dx)2

xcos

2

x(sin

19.

2

0

2dx

4x

3x620.

1

0

x dx)4

xsinxe(


21.

3

1

2x1

dx equals

(a)3

(b)

3

2(c)

6

(d)

12

22.

3

2

0

2x94

dx equals

(a)6

(b)

12

(c)

24

(d)

4

MI – 16


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Answers :

1. 2 2.2

3log 3.

3

644.

2

15. 0

6. e4 (e – 1) 7. 2log2

18.

32

12log 9.

2

10.

4

11.2

3log

2

112.

4

13. 2log

2

114. 5tan

5

36log

5

1 1

15. )1e(2

1 16.

2

3log

4

5log9

2

55 17. 2

21024

4

18. 0 19.8

32log3

20.

2241

21. d 22. c

NCERT Solved examples upto the section 7.9 (Evaluation of Definite Integrals by Substitution) :

Example 28 : Evaluate

1

1

54 dx1xx5 .

Solution : 3

24

Example 29. : Evaluate

1

0

2

1

dxx1

xtan.

Solution : 32

2

EXERCISE 7.10

Evaluate the following integrals by using substitution.

1.

1

0

2dx

1x

x2.

2

0

5 dcossin 3.

1

0

2

1 dxx1

x2sin

4.

2

0

2xx (Put x + 2 = t2) 5.

2

0

2dx

xcos1

xsin

6.

2

0

2x4x

dx7.

1

1

2 5x2x

dx8.

2

1

x2

2dxe

x2

1

x

1

MI – 17


New Delhi – 110 018, Ph. : 9312629035, 8527112111


9. The value of the integral

1

3

14

3

13

dxx

)xx( is

(a) 6 (b) 0 (c) 3 (d) 4

10. If

x

0

)x(fthen,tdtsint)x(f is

(a) cosx + x sin x (b) x sin x

(c) x cos x (d) sin x + x cos x

Answers :

1. 2log2

12.

231

643. 2log

2

4. )12(15

216 5.

4

6.

4

17521log

17

1

7.8

8.

4

)2e(e 22 9. d

10. b

NCERT Solved examples upto the section 7.10 (Some Properties of Definite Integrals) :

Example 30 : Evaluate dxxx

2

1

3

.

Solution : 4

11


4

4

2 dxxsin

Solution : 2

1

4


0

2dx

xcos1

xsinx.

Solution : 4

2


1

1

45 dxxcosxsin .

Solution : 0

MI – 18


New Delhi – 110 018, Ph. : 9312629035, 8527112111


2

0

44

4

dxxcosxsin

xsin.

Solution : 4


3

6

xtan1

dx.

Solution : 12


2

0

dxxsinlog .

Solution : 2log2

EXERCISE 7.11

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

1.

2

0

2 dxxcos 2.

2

0

dxxcosxsin

xsin3.

2

0 2

3

2

3

2

3

xcossin

xdxsin

4.

2

0

55

5

xcosxsin

dxxcos5.

5

5

dx|2x| 6.

8

2

dx|5x|

7.

1

0

n dx)x1(x 8.

4

0

dx)xtan1log( 9.

2

0

dxx2x

10.

2

0

dx)x2sinlogxsinlog2( 11.

2

2

2 dxxsin

12.

0

xsin1

xdx13.

2

2

7 dxxsin 14. 2

0

5 dxxcos

MI – 19


New Delhi – 110 018, Ph. : 9312629035, 8527112111

15

2

0

dxxcosxsin1

xcosxsin16.

0

dx)xcos1log( 17.

a

0

dxxax

x

18.

4

0

dx|1x|

19. Show that

a

0

a

0

dx)x(f2dx)x(g)x(f , if f and g are defined as f(x) = f(a – x) and g(x) + g(a – x) = 4


20. The value of

2

2

53 dx)1xtanxcosxx( is

(a) 0 (b) 2 (c) (d) 1

21. The value of

2

0

dxxcos34

xsin34log is

(a) 2 (b)4

3(c) 0 (d) –2

Answers :

1.4

2.

4

3.

4

4.4

5. 29 6. 9

7. )2n)(1n(

1

8. 2log

8

9.

15

216

10.2

1log

2

11.

2

12.

13. 0 14. 0 15. 0

16. – log 2 17.2

a18. 5

20. c 21. c

MISCELLANEOUS EXAMPLES :

Example 37 : Find dxx6sin1x6cos .

Solution : C)x6sin1(9

12

3

MI – 20


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Example 38 : Find

dxx

)xx(5

4

14

.

Solution : Cx

11

15

4 4

5

3

Example 39 : Find )1x)(1x(

dxx2

4

.

Solution : Cxtan2

1)1xlog(

4

1|1x|log

2

1x

2

x 122

Example 40 : Find

dx

)x(log

1)xlog(log

2.

Solution : Cxlog

x)xlog(logx

Example 41 : Find dxxtanxcot .

Solution : Cxtan2

1xtantan2 1

Example 42 : Find )x2(cos9

xdx2cosx2sin

4.

Solution : Cx2cos3

1sin

4

1 21


2

3

1

dx|)xsin(x| .

Solution : 2

13


0

2222 xsinbxcosa

xdx.

Solution : ab2

2

MI – 21


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MISCELLANEOUS EXERCISE ON CHAPTER 7


1.3xx

1

2.

bxax

1

3.

2xaxx

1

[Hint : Put x = a/t]

4.

4

342 )1x(x

1

5.

3

1

2

1

xx

1

6.)9x)(1x(

x52

7.)axsin(

xsin

8. xlog2xlog3

xlog4xlog5

ee

ee

9.

xsin4

xcos

2

10.xcosxsin21

xcossin22

88

11.

)bxcos()axcos(

1

12.8

3

x1

x

13.

)e2)(e1(

exx

x

14.

)4x)(1x(

122

15. cos3x elog sin x 16. e3 logx (x4 + 1)–1 17. n)]bax(f)[bax(f

18.)xsin(xsin

1

3 19. ]1,0[x,

xcosxsin

xcosxsin11

11

20.x1

x1

21.

xex2cos1

x2sin2

22.

)2x()1x(

1xx2

2

23.x1

x1tan 1

24.

4

22

x

xlog2)1xlog(1x

Evaluate the definite integrals in Exercises 25 to 33.

25.

2

x dxxcos1

xsin1e 26.

4

0

44 xsinxcos

xcosxsin27.

2

0

22

2

xsin4xcos

xdxcos

28.

3

6

dxx2sin

xcosxsin29.

1

0xx1

dx30.

4

0

dxx2sin169

xcosxsin

31.

2

0

1 dx)x(sintanx2sin 32.

0

dxxtanxsec

xtanx33.

4

1

dx|]3x||2x||1x[

Prove the following (Exercises 34 to 39)

34.

3

1

2 3

2log

3

2

)1x(x

dx35.

1

0

x 1dxxe 36.

1

1

417 0xdxcosx

MI – 22


New Delhi – 110 018, Ph. : 9312629035, 8527112111

37.

2

0

3

3

2xdxsin 38.

4

0

3 2log1xdxtan2 39.

1

0

1 12

xdxsin

40. Evaluate

1

0

x32 dxe as a limit of a sum.

Choose the correct answer in Exercises 41 to 44.

41. xx ee

dx is equal to

(a) tan–1(ex) + C (b) tan–1 (e–x) + C

(c) log (ex – e–x) + C (d) log (ex + e–x) + C

42. dx

)xcosx(sin

x2cos2

is equal to

(a) Cxcosxsin

1

(b) log |sin x + cos x| + C

(c) log |sin x – cos x| + C (d)2)xcosx(sin

1

43. If f (a + b – x) = f(x), then b

a

dx)x(xf is equal to

(a)

b

a

dx)xb(f2

ba(b)

b

a

dx)xb(f2

ba

(c)

b

a

dx)x(f2

ab(d)

b

a

dx)x(f2

ba

44. The value of

1

0

2

1 dxxx1

1x2tan is

(a) 1 (b) 0 (c) –1 (d)4

Answers :

1. Cx1

xlog

2

12

2

2. C)bx()ax()ba(3

22

3

2

3

3. Cx

)xa(

a

2

4. C

x

11

4

1

4

5. C)x1log(6x6x3x2 6

1

6

1

3

1

6. C3

xtan

2

3)9xlog(

4

1|1x|log

2

1 12

MI – 23


New Delhi – 110 018, Ph. : 9312629035, 8527112111

7. sin a log|sin (x – a)| + x cos a + C 8. C3

x3

9. C2

xsinsin 1

10. Cx2sin2

1

11. C)axcos(

)bxcos(log

)basin(

1

12. C)x(sin

4

1 41

13. Ce2

e1log

x

x

14. C

2

xtan

6

1xtan

3

1 11

15. Cxcos4

1 4 16. C)1xlog(4

1 4

17. C)1n(a

)]bax(f[ 1n

18. Cxsin

)xsin(

sin

2

19. Cxxx2

xsin)1x2(2 2

1

20. Cxxxcosx12 21

21. ex tan x + C 22. C|2x|log31x

1|1x|log2

23. Cx1xcosx2

1 21

24. C

3

2

x

11log

x

11

3

12

2

3

2

25. 2e

26.8

27.

6

28.

2

)13(sin2 1

29.3

2430. 9log

40

131. 1

2

32. )2(

2

33.2

19

40.

e

1e

3

1 2

41. a 42. b 43. d 44. b

Documents

INTEGRALS - einstein classeseinsteinclasses.com/Bluetooth Folder/Integeration.pdf · m 15. 9 4x2 x 16. e2x + 3 ... 1 sin2x cosx sinx ... NCERT Solved examples upto the section 7.4