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The value of int0 ^(pi/2) (cos3x+1)/(cos...

The value of `int_0 ^(pi/2) (cos3x+1)/(cosx - 1) dx` is equal to

A

2

B

1

C

`1/2`

D

0

Text Solution

Verified by Experts

The correct Answer is:
B
Let `l = int_(0)^(pi//2)(cos 3x+1)/(2 cos x-1)dx`
`= int_(0)^(pi//2)(cos 3x-cos.(3pi)/(3))/(2(cos x-cos.(pi)/(3)))dx`
`= int_(0)^(pi//2)((4 cos^(3)x-3cos x)-(4 cos^(3)(pi)/(3)-3cos.(pi)/(3)))/(2(cos x-cos.(pi)/(3)))dx`
`= 2int_(0)^(pi//2)((cos^(3)x-"cos"^(3)(pi)/(3))/(cos x-cos.(pi)/(3)))dx-(3)/(2)int_(0)^(pi//2)((cos x - cos.(pi)/(2))/(cos x-cos.(pi)/(3)))dx`
`= 2int_(0)^(pi//2)(cos^(2)x + "cos"^(2)(pi)/(3)+cos x cos.(pi)/(3))dx-(3)/(2)int_(0)^(pi//2)2dx`
`= int_(0)^(pi//2)(1+cos 2x+(1)/(2)+cos x)dx-(3pi)/(4)`
`= (3pi)/(4)+1-(3pi)/(4)=1`
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