`int_0^oo(dx)/([x+sqrt(x^2+1)]^3)i se q u a lto` `3/8` (b) `1/8` (c) `-3/8` (d) none of these

Options

A. `3/8`
B. `1/8`
C. `-3/8`
D. none of these

Answer

A

Explanation

Putting `x=tan theta` we get ,
`int_(0)^(pi//2) (dx)/([x+sqrt(x^(2)+1]^(3)))=int_(0)^(pi/2) (sec^(2) theta d theta)/((tan theta +sec theta)^(3))`
`=int_(0)^(pi//2) (cos theta)/((1+sin theta)^(3))d theta`
`=[-1/(2(1+sin theta)^(2))]_(0)^(pi//2)=-1/8+1/2=3/8`