If `f(x)="cos"(tan^(-1)x),` then the value of the integral `int_0^1xf^(x)dx"i s"` `(3-sqrt(2))/2` (b) `(3+sqrt(2))/2` 1 (d) `1-3/(2sqrt(2))`

Options

A. `(3-sqrt(2))/2`
B. `(3+sqrt(2))/2`
C. `1`
D. `1-3/(2sqrt(2))`

Answer

D

Explanation

`f(x)=cos(tan^(-1)x)`
`:.f'(x)=-(sin (tan^(-1)x))/(1+x^(2))`
`:. I=int_(0)^(1)x f''(x)dx`
`=[f'(x)]_(0)^(1)-int_(0)^(1)f'(x)dx` (Integrating by parts)
`=[f'(1)]-[f(x)]_(0)^(1)`
`=f'(1)-f(1)+f(0)`
Now `f(0)=1,f'(1)=-1/(2sqrt(2)),f(1)=1/(sqrt(2))`
`:. I=1-3/(2sqrt(2))`