TOPIC – 16: DEFINITE INTEGRALS

PROVE THAT:
 /2  /2 
 sin 5 / 3 x  
 (  (1 + cos sin x) dx = sin 
cos x x
dx = dx =
1.
0
(sin x + cos x ) 4
18.
0
sin x + cos x
5/3 5/3
4 ) 35.
0
 /2  /2  /2
 (sin x − cos x) 
 ( sin x +  (1 + sin x cos x) dx = 0  log (sin 2x)dx = − 2 (log 2)
sin x
2.
0
cos x ) dx =
4
19.
0
36.
0
 /2 1 
3
 2
( 
20. x(1 − x )5 dx =

37. x log (sin x )dx = − (log 2)
sin x 1
dx =
3.
0
sin x + cos3 x
3
4 ) 0
42
0
2
 /2 4  /2
4
 
 (sin 
21. x(4 − x )3 / 2 dx =
 (2 log cos x − log sin 2x)dx = − 2 (log 2)
cos x 512
dx =
4.
0
4
x + cos4 x 4 ) 0
35
38.
0
 /4  
n

 (sin  
39. log (1 + cos x )dx = − (log 2)
sin x
dx = 22. sin 2 x cos3 xdx = 0
5.
0
n
x + cos x
n
4 ) 0 0
 /2   /2
3/ 2
  2

 (sin   log (tan x + cot x)dx =  (log 2)

sin x
dx = 23. x cos2 xdx =
6.
0
3/ 2
x + cos x
3/ 2
4 ) 0
4
40.
0
 /2  
1/ 4
     1
 (sin  (1 + x)(1 + x ) dx = 4 1+ x  log  x + dx =  (log 2)
cos x x 1
dx =
7.
0
1/ 4
x + cos1 / 4 x 4 ) 24.
0
2
41. 
0
2
  x
 /2 a  /2
  
 ( tan x +  x+  x cot xdx = 2 (log 2)
tan x dx
8.
0
cot x ) dx = 4 25.
0 a −x
2 2
=
4
42.
0
 /2 a 1
   sin −1 x 
dx =  (log 2)
 ( tan x + ( 
cot x x
43. 
9.
0
cot x ) dx =
4
26.
0
x + a−x ) dx =
4
0
 x  2
 /2 1 1
 1  
 (1 + tan x) = 4   (log 2)
dx log x
10. 3
27. log  − 1dx = 0 44. dx = −
0 0
x  0 1− x 2 2
 /2  /2
log (1 + x )
( )
1
 cos2 x 
   (1 + x ) dx = 8 (log 2)
dx 1
11. = 28. dx = log 2 + 1 45.
0
(1 + tan x ) 4 0
(sin x + cos x ) 2 0
2

 /2  
  2

  
dx x tan x
12. = 29. dx = 46. x12 sin 9 xdx = 0
0
(1 + cot x ) 4 0
sec x cos ecx 4
−
 /2  a
 2
  x
x tan x
13. sin 2 xdx = 30. dx = 47. 3
a 2 − x 2 dx = 0
4 sec x + cos x 4
0 0 −a
 /2  /2  

 (sin )
 
 cos xdx =  sin  1 + sin x dx =   2 − 1
x sin x
14. n n
xdx 31. 48. 75
x + x125 dx = 0
0 0 0 −
 /2  1
 2
 (1 +  
49. e x dx = 2(e − 1)
tan x x
15.
0
tan x ) dx =
4
32.
0
1 + sin x
2
dx =
2 2 −1
 /2  2 x + 1, when1  x  2 
  
50. let f (x ) = 
 (1 + 
cot x
sin 2m x cos 2m +1 xdx = 0
16.
0
cot x ) dx =
4
33.
0
 x + 1, when 2  x  3

2 

3
 /2  /2
 f (x)dx =
 34
 (1 +
dx
 (sin x − cos x)log (sin x + cos x)dx = 0
show that .
17.
0
tan x ) =
4
34.
0
1
3

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 π/2 π/2
π
51.  log ( sinx ) dx=  log ( cosx ) dx=- 2 ( log2 )
0 0
π/2

( )
2
sin x 1
52. 0 ( sinx+cosx )
dx=
2
log 2+1

3π/10
sinx π
53.  ( sinx+cosx ) dx= 20
π
5
5
54.  x + 2 dx=29
-5

1
13
55.  5 x − 3 dx=
0
10

π
55.  cosx dx=2
0
4
56.  ( x − 1 + x − 2 + x − 3 ) dx =
19
2
1

4
57.  ( x − 1 + x − 2 + x − 3 ) dx =
19
2
1

 f ( x ) dx,where f(x) is defined by

0

 π
 sinx,if0£x£
2  π 4
58. Evaluate:  5- 2 +e 
 π
f ( x ) = 1,if £x£5
 2
e x-5 ,if5£x£9


π
x π2
59.  2
dx=
0
cos x 2 2

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 1 
1
60. 0  x − 1 dx=0
log

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