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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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) :
Example 8 : Find the following integrals :
(i) 16x
dx2
(ii) 2xx2
dx
Solution : (i) C4x
4xlog
8
1
(ii) sin–1(x – 1) + C
Example 9 : Find the following integrals :
(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
Example 10 : Find the following integrals :
(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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
EXERCISE 7.7
Integrate the functions in Exercises 1 to 9.
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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(
Choose the correct answer in Exercises 21 and 22.
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
Choose the correct answer in Exercises 9 and 10.
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
Example 31 : Evaluate
4
4
2 dxxsin
Solution : 2
1
4
Example 32 : Evaluate
0
2dx
xcos1
xsinx.
Solution : 4
2
Example 33 : Evaluate
1
1
45 dxxcosxsin .
Solution : 0
MI – 18
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
Example 34 : Evaluate
2
0
44
4
dxxcosxsin
xsin.
Solution : 4
Example 35 : Evaluate
3
6
xtan1
dx.
Solution : 12
Example 36 : Evaluate
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Choose the correct answer in Exercises 20 and 21.
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Example 43 : Evaluate
2
3
1
dx|)xsin(x| .
Solution : 2
13
Example 44 : Evaluate
0
2222 xsinbxcosa
xdx.
Solution : ab2
2
MI – 21
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MISCELLANEOUS EXERCISE ON CHAPTER 7
Integrate the functions in Exercises 1 to 24.
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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