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The value of the definite integral int(2...

The value of the definite integral `int_(2pi)^(5pi//2)(sin^(-1)(cosx)+cos^(-1)(sinx))dx` is equal to

A

`pi^2/8`

B

`pi^2/4`

C

`pi^2/2`

D

`pi^2`

Text Solution

Verified by Experts

The correct Answer is:
B
`:. sin^(-1) (cos x ) + cos^(-1) (sinx)` is periodic with period `2pi`, then
`int_(2pi)^(5pi//2) (sin^(-1) (cosx) + cos^(-1)(sinx))dx`
`=int_(0)^(pi//2) (sin^(-1) {cos(pi/2-2)} + cos^(-1){sin(pi/2-x)})dx`
`=int_(0)^(pi//2) (sin^(-1)sinx + cos^(-1)cosx)dx`
`=2int_0^(pi//2) xdx = 2{x^2/2}_0^(pi/2) =pi^2/4`
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