ZERO LEVEL Definite Integrals-I - IMPETUS EDUCATIONimpetuseducation.com/zero_level/Definite Integrals -1,2.pdf · 3 x2 4 dx ZERO LEVEL Definite Integrals-I 29. ... x sinx sin2x dx

18, Zone-II, M.P. Nagar, Bhopal : 0755-2768319 Class-Sheet |1

Evaluate the following integrals:

1. 9

4dxx 2.

1

010 dxx

3. 1

1

3 dxx 4. 3

1

4 dxx

5. 2

1

5 dxx 6. 16

0

4/3 dxx

7.

1

4 xdx

8. 8

1 3/2xdx

9. 1

0 3 xdx

10. 4

1

2/7 dxx

11. 2/

0

2cos dxx 12. 2/

0

2tan dxx

13. 2/

0tan dxx 14.

2/

0

2sin dxx

15.

2/

4/

2cosec dxx 16.

4/

6/cosec dxx

17. 3/

0

3cos dxx

18. (i) 2/

0

3cos dxx (ii) 2/

0

3sin dxx

19. 2/

0

4sin dxx 20. 2/

02sinsin dxxx

21.

03cos2sin dxxx 22.

2/1

0 1 xdx

23.

1

0 21 x

dx

24. (i)

0 21 xdx

(ii)

)9( 2xdx

25.

1

0 11

dxxx

26.

2

0 2 1xx

dx

27.

a

xax

dx0 2 28.

4

3 2 4xdx

ZERO LEVEL Definite Integrals-I

29.

2

0 24 xxdx

30.

a

xaxadx

0 22

31.

4

0 2 32xx

dx32.

2

0 2 34xx

dx

33.

2

1 )2)(1( xxdx

34. 1

0)1( dxxx

35. (i)

4/

02sin1 dxx (ii)

2/

02cos1 dxx

36. (i)

2/

0

22 )tancos( dxxbxa

(ii)

4/

3/

2)cot(tan dxxx

37. (i)

2/

0cos1 dxx (ii)

2/

0sin1 dxx

38. (i)

4/

0 )2cos1( xdx

(ii)

2/

4/ )2cos1( xdx

(iii)

4/

4/ )sin1( xdx

39.

0sinlog2cos dxxx 40.

2/

0

2 cos dxxx

41. (i) 4/

0

2 sin dxxx (ii) 2/

0

2 2cos dxxx

42. 2/

0

22 cos dxxx 43. 2/

0

3 3sin dxxx

44. 2/

02sinsin dxxxx 45.

3

1 2)1(log

dxxx

46.

2

1 )1( xxdx

47. 2

1 )2()3(

dxxx

x

48. 2

1 )2()3(

dxxx

x49.

3

1 2 )1(xxdx

50. 1

0dxxex

51.

1

0 2 )1(dx

xxex

52.

2

0 2)(log1

log1e

dxxx

53.

2

1 2

2

)34(5

dxxx

x54.

1

0 2 42)3(

dxxx

x

55. /2

0

3tan2 dxx

07/IX/27/ZLCS-01/A/Ex18.1

2| Class-Sheet 18, Zone-II, M.P. Nagar, Bhopal : 0755-2768319

ZERO LEVEL Definite Integrals-II

34. (i)

2/

0 )sin45( xdx

(ii)

0 )cos6( xdx

(iii)

0 )cos45( xdx

35. (i)

2/

0 )sin2(cos xxdx

(ii)

2/

0 )sincos3( xxdx

36. (i)

4/

0

3

)2cos1(tan

dxx

x

(ii)

2/

0 2 )2cos3(coscossin

dxxx

xx

37.

2/

0 44 )cos(sin2sin

dxxx

x38.

2/

3/ 2/5)cos1(cos1

dxx

x

39. 1

0)1log( dxx 40.

2

1

2 log dxxx

41. 1

0

1sin dxx 42. 1

0

1sin dxx

43. (i) 1

0

21 )(cos dxx (ii) dxxx 1

0

21 )(tan

44. dxxx

1

0 21

12

sin 45.

1

0 2

21

11

cos dxxx

46.

2

1 2)log1( xxdx

47. dxxxx

2/

6/ 2cosec1cotcosec

48. dxx 4/

0cot

49. dxxx )cottan(2/

0

50. dxxx

x

2/

0 )2/sin2/(coscos

51. dxxx 1

0

43 31 52. dxxx

1

0 22

2

)1()1(

53.

2

1 2 1)1( xx

dx54.

15

8 1)3( xxdx

55.

2

1 )2)(1( xxdx

56.

3

2 265

)2(dx

xx

x

57.

16

0 4/1

4/1

1dx

xx

1. 4

012 dxx 2.

1

0 45xdx

3. a

dxbax0

6)( 4. 2

0

2/ dxex

5. 5/1

0

3510 dxx 6.

4

2 2 1dx

xx

7.

1

0 )32( xdx

8.

1

0 412

dxxx

9. 2

1 2

/1dx

xe x

10. 1

0

2dxxex

11.

1

0 2 )15(5

dxx

x12.

2

1 2 193

dxx

x

13.

7

2

2

2dx

xx

14.

2

1

2

1dx

xx

15.

1

0 21dx

ee

x

x

16.

1

0 xx eedx

17. e

e xxdx

/1 3/1)(log 18.

6/

0 )sin43(cos

dxx

x

19.

2/

0 cos1sin

dxx

x20.

2/

0

3sincos dxxx

21.

1

0 4 )sin1(cossin

dxxxx

22.

1

0 2

1

1tan

dxx

x

23.

1

0 2

1

1tan

dxx

x24.

1

0 2/32

1

)1(tan

dxx

xx

25. 2

123 dxxx 26.

2

02 dxxx

27. a

dxxa0

22 28. 2

0

22 x

29.

adx

xa

x0 22 30.

a

xdx

0 1

31. (i)

2/

0cos1 dxx (ii)

2/

0sin1 dxx

32. (i)

2/

0 2 )cos1( xdx

(ii)

2/

0 2 )cos94( xdx

33. (i)

2/

0 2222 )sincos( xbxadx

(ii)

2/

0 2222 )sincos(2sin

xbxadxx

07/X/03/ZLCS-02/A/Ex18.2


ZERO LEVEL Definite Integrals-III

15. (i)

2/

0 3 4)tan1(dx

xdx

(ii)

2/

0 3 4)cot1(dx

xdx

16.

1

001

1log dx

x

17.

1

0 2 2log8)1(

)1log(dx

xx

18.

2/

0

2

332

)cossin1(sin

dxxx

x

19.

2/

0

2)12log(

21

)cos(sincos

dxxx

x

20. (i)

0

22

4cos dxxx

(ii)

0

2

4)cosec(sectan

dxxx

xx

21.

0

2

4)cos(sectan

dxxx

xx

22.

01

2)sin1(sin

dxx

xx

23.

0

2

2 22)sin1(dx

xx

24.

2/

02log

2)2sinlogsinlog2( dxxx

25.

2/

00)coslog(sin)cos(sin dxxxxx

26.

2/

02log

2)2log(sin x

27.

0

22log

2)log(sin dxxx

28.

0

2log)cos1log( dxx

29.

2/

02log)cotlog(tan dxxx

1.

2/

0 4)cos(sincos

dxxx

x

2.

2/

0 4cossinsin

dxxx

x

3.

2/

0 4cossinsin

dxxx

xnn

n

4.

2/

0 4cossincos

dxxx

xnn

n

5.

2/

0 33

3

4cossincos

dxxx

x

6. (i)

2/

0 44

4

4)cos(sincos

dxxx

x

(ii)

2/

0 44

4

4)cos(sinsin

dxxx

x

7. (i)

2/

0 4cottantan

dxxx

x

(ii)

2/

0 4cottancot

dxxx

x

8.

2/

0 4/14/1

4/1

4)cos(sincos

dxxx

x

9.

2/

0 2/32/3

2/3

4)cos(sinsin

dxxx

x

10.

2/

0 77

7

4)cos(sinsin

dxxx

x

11.

2/

0 55

5

4)cos(sinsin

dxxx

x

12.

2/

0 4)tan1( xdx

13.

2/

0 4)cot1( xdx

14. (i)

2/

0 4)tan1(tan

dxx

x

(ii)

2/

0 4)cot1(cot

dxx

x

07/X/05/ZLCS-03/A/Ex18.3&18.4


30.

0

122 0cossin xdxx mm , where m is a posit ive

integer.

31.

0

32 0cossin dxxx

32. 4

0

2/335512

)4( dxxx

33.

1

0 )2)(1(1

)1(nn

dxxx n

34.

a adx

xaxx

0 2)(

35.

adx

xax

dx0 22 4)(

36.

0 2 4)1)(1(

dxxx

x

37. (i)

4/3

4/ 4)(

a

a

adx

xaxx

(ii)

4

1 23

)5(dx

xxx

38. (i)

4/3

4/2

)cos1( xdx

(ii)

4/3

4/)12(

)sin1(dx

xx

39.

8/3

8/ 8)sin(coscos

dxxx

x

40.

2/

02log

2cot dxxx

41.

1

0 22log

21

logdx

x

x

42.

1

0

12log

2sin

dxx

x

43. a

adxxax 0223 44.

0sin912 dxxx

45.

2/

2/

7 0dxxx

46. (i) 2

1 25

|| dxx (ii) 1

2 29

|12| dxx

47. 8

017|5| dxx

48. (i)

2

04|cos| dxx

(ii)

4/

4/)22(|sin| dxx

49. Let

32when,121when,12

)( 2 xxxx

xf , show that

3

1 334

)( dxxf

50. Evalute 4

1)( dxxf , where

42if,2321if,34)(

xxxxxf

51. Let

42when,2920when,43)(

2

xxxxxf show that

4

066)(xf

Evaluate the following:

1. 5

0)1( dxx 2.

2

1)23( dxx

3. 2

0)72( dxx 4.

3

1

2 dxx

5. 3

0

2 )2( dxxx 6. 2

0

2 )43( dxx

7. 2

0

2 )1( dxx 8. 2

0

2 )( dxxx

9. 4

1

2 )23( dxxx 10. 3

1

2 )52( dxxx

11. 1

0

2 )52( dxxx 12. 1

0

2 )123( dxxx

13. 4

2

2 )23( dxxx 14. 2

0

3 )1( dxx

15. 2

0dxex 16.

3

1dxe x

17. 1

0|13| dxx 18.

2

0|1| dxx


Answers 18.1

1. 338

2. )10(log9

3. 0

4. 5242

5. 6415

6. 7512

7. 4log 8. 3 9. 3/2

10. 8033

11. 4

12.

41

13. log 2 14. 4

15. – 2

16. |)32)(12(|log 17.833

18. (i) 32

(ii) 32

19. 163

20. 32

21. 54

22. )22(

23. 2

24. (i) 2

(ii) 3

25. )12log2( 26. 4

27.

28.

35

log41

29. 417521

log17

1

30. 2537

log5

1

a

31. |31|log|275|log

32. |83|log|154|log

33. 34. 8

35. (i) 1 (ii) 2

36. (i) )(4

ba

(ii) 3

2

37. (i) 2 (ii) 2

38. (i) 21

(ii) 21

(iii) 2

39. 2

40.

2

4

2

41. (i)

2

26222

2

(ii) 4

42.

848

3

43.

12272 2

44.

21

4 45. 2log3log43

46. )3log4(log 47. 6log21

48.

152

5log6log 49.

3

1 2 )1(xxdx

50. 1 51.

1

2e

52.

e

e2

2

53.

23

log45

log925

5

54.

3

2tan

63

4)4log7(log

21 1

55. )2log1(

Exercise 18.2

1. 326

2. 52

3.a

bba7

)( 772 4. )1(2 e

5. 10log9000

6. ]5log17[log21

7.

31

log21

8.4

9.eeee

10.

21e

11. 6log51

12. )8log35(log61

13. 15626

14. 1556

15.

4tan 1 e 16.

4tan 1

e

17. 0 18. )3log5(log41

19. )12(2 20. 218

21. 8

22.32

2

23.12

24.8

)4(2

25. 514

26.15

216


27.4

2a28. 2

29. )12( a 30. )2log46( 31. (i) 2 (ii) 2

32. (i) 22

(ii) 134

33. (i) ab2

(ii) 22

1ba

34. (i)

31

tan32 1

(ii) 35

(iii) 3

35. (i) 253

log5

1

(ii) 110

110log

210

210log

10

1

36. (i) 81

(ii) )8log9(log

37. 2

38. 23

39. 12log2 40. 97

2log38

41.

1

2 42. 4

43. (i) )2( (ii)

1

44

44.

2log

2 45. 2log2

46. 2log12log

47.

2log

2

48. )12log(2

1

22

49. 2 50. )12(2

51. 187

52. 21

53. 3

154. ]3log5[log

21

55. )8log9(log 56. 2

57. 81log3

16

Answers : 18.4

1. 235

2. 25

3. 18 4. 326

5. 18 6. 0

7. 314

8. 314

9. 78 10. 3112

11. 611 12. 3

13. 314

14. 6

15. )1( 2 e 16.

3

2 1e

e

17. 1 18. 1

Documents

ZERO LEVEL Definite Integrals-I - IMPETUS EDUCATIONimpetuseducation.com/zero_level/Definite Integrals -1,2.pdf · 3 x2 4 dx ZERO LEVEL Definite Integrals-I 29. ... x sinx sin2x dx