TOPIC – 16: DEFINITE INTEGRALS

EVALUATE THE FOLLOWING INTEGRALS:

−1
dx dx dx dx
      
3 2 16 3 4 1 8
1. (i) x 4dx (ii) (iii) x −5dx (iv) x 4dx (v) (vi) (vii)
−4 x
2
1 1 0 1 x 0
3
x 1
x 3

  
dx
    
3 7
6 4 2
2. (i) 5x + 6 dx (ii) 3. (i) sec2 xdx (ii) cos ec2 xdx (iii) cot2 xdx
−1 0 3x + 4 0 −
4

4
   

   
4 3 4 2
(iv) tan2 xdx 4. (i) tan xdx (ii) cos ecxdx 5. (i) sin2 xdx (ii)
0 0  0
6



4
cos2 xdx
0
   

   
2 6 3
6. (i) sin x. sin 2x.dx (ii) cos x. cos 2x.dx (iii) sin 2x. cos 3x.dx (iv) cos3 x.dx
0 0 0 0

dx 
dx  4 dx  2 dx  /4
7. (i)     
1
(ii) 8. (i) (ii) 9. (i) 1 − sin 2x .dx
0 1 − x2 0 (1 + x 2 ) 0 (1 + cos 2x )  4 (1 − cos 2 x ) 0

 /2  /2  /2  /2
(ii) 0
1 − cos 2x .dx 10. (i) 
0
sin3 x.dx (ii)  0
cos4 x.dx 11. 
0
(a cos2 x + b sin2 x ).dx
 /4  /4  /2  /2
dx
12.  (tan x + cot x )2 dx 13.  14.  1 + sin x .dx 15.  1 + cos x .dx
 /3 − / 4 (1 + sin x ) 0 0

dx dx dx dx dx
  (1 + x + 2x )   
4 1 2 12 4
16. 17. 18. 19. 20.
3 (x − 4)
2
0
2
0 (4 + x − x 2 ) 14 x−x 2 0 x + 2x + 3
2

a
dx dx dx dx
    
2 2 1 2
21. 22. 23. 24. x.(1 − x )dx 25.
1 x + 4x + 3
2 1 (x − 1).( 2 − x ) 0 (ax + a 2 − x 2 ) 0 1 x.(1 + x )
(x + 3) dx dx  2

    
2 3 2 1
26. dx 27. 28. 29. x.exdx 30. x 2. cos x.dx
1 x.( x + 2) 1 x .( x + 1)
2
1 x.(1 + 2x )2 0 0
 4  2 log x  2

  33.  log x.dx  (1 + x)  x. sin x. sin 2x.dx

2 3
31. x 2 sin x.dx 32. x 2 cos 2x.dx 34. 2
dx 35.
0 0 1 1 0

e2  1 1  e
 1 + x log x   2  2
36. 0
 − 2
 log x (log x ) 
dx 37. 1
ex 
 x
dx

38. 0
x 2 cos2 x.dx 39. 
0
x 3 sin 3x.dx

 6 5x 2 (x − 3) xe x  sin x
   (x  (1 + x ) dx 44. 
2 1 1
40. (2 + 3x 2 ) cos 3x.dx 41. dx 42. dx 43. dx
0 1 (x + 4 x + 3)
2
0
2
+ 2 x + 4) 0
2
0 (sin x + cos x )

dx π
0 (1+x+x 2 ) 3 3
1
45. =

dx 3 1
 ( ) ( log5)
2
= log2 -
x (1+x 2 ) 2
46.
1 2

π
8
47.  0
2
cosθsin 3θdθ=
21
π
dx π
48.  0
2
=
( 4sin 2 x+5cos2 x ) 4 5
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 xtan -1x 2 ( 4-π )

1
49. 3
dx=
(1+x ) 8
0
2 2

1
sin -1x π 1
50.  2
3
dx= - log2
(1-x ) 4 2
0
2 2


51. 0
2

 x x
cosx
3
dx= 2 ( )
2 −1
 cos + sin 
 2 2

cosx π 1 
52. 02 (1+cosx+sinx ) dx=  4 - 2 log2 

a-x

a
53. dx=aπ
-a a+x

1-x 2 π 1
0 1+x 2  4 - 2 
1
54. x dx=

π
cosx  3π log3 
55.  ( 3cosx+sinx ) dx=  20 -
0
2
10 

π
56.  (
0
2
)
tanx + cotx dx= 2π

x π 

a
57. sin -1 dx=a  -1
0 a+x 2 

π

1
58. sin -1 xdx=
0 4
ππ 
0 ( )
1 2
-1
59. x tan x dx=  -1
4 4 
π
cosecxcotxdx -1 1
60. 6 1+cosec2 x
π
2
=tan
3

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 ANSWERS
242 15 512 3
1. (i) (ii) − log 4 (iii) (iv) (v) 2 (vi)
5 64 7 2
(vii) 3
42
2. (i) (ii) 2
5
1    
3. (i) (ii) – 2 (iii) 1 −  (iv) 1 − 
3  4  4
4. (i) log 2 (ii) log ( )(
2 −1 2 + 3 )
  1
5. (i) (ii)  + 
4  8 4
2 5 4 3 3
6. (i) (ii) (iii) − (iv)
3 12 5 8
 
7. (i) (ii)
2 2
1 1
8. (i) (ii)
2 2
9. (i) 2 −1( ) (ii) 2
2 3
10. (i) (ii)
3 16

11. ( a + b)
4
2
12. −
3
13. 2
14. 2
15. 2
1 5
16. log
4 3
2  −1 5 1 
17.  tan − tan −1 
7 7 7
1 21 + 5 17
18. log
17 4

19.
6
(
20. log 2 + 3 )
21. log 4 + 15 + log 3 − 8
22. 
1 7+3 5
23. log
5a 2
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 
24.
8
25. ( log 4 − log 3)
1
26. log 6
2
2 2
27. + log
3 3
 2
28.  log 6 − log 5 − 
 15 
29. 1
2 
30.  − 2
 4 
  2 
31.  2 + − − 2
 2 2 16 2 

32. −
4
33. ( 2 log 2 ) − 1
3
34. log 3 − log 2
4
 1
35.  − 
 4 2
 e2 
36.  − e 
2 
37. ee
2  
38.  − 
 48 8 
 2 2 
39.  − 
 27 12 
  2 + 16 
40.  
 36 
5 5 3
41. 5 −  9 log − log 
2 4 2
1 4  2 
42. ( log 7 − log 4 ) +  − tan −1 
2 36 3
e 
43.  − 1
2 

44.
2
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