` int _(0)^(oo) (dx)/((x+ sqrt(x^(2)+1))^(3))` is equal to

Options

A. `3/8`
B. `1/8`
C. `(-3)/8`
D. None of these

Answer

A

Explanation

Let `l =int_(0)^(infty)(dx)/((x+sqrt(x^(2)+1))^(3))`
Put `x=tanthetarArrdx=sec^(2) thetad theta`
`thereforel =int_(0)^(pi//2)(sec^(2)thetad theta)/((tan theta+sqrt(tan^(2) theta+1))^(3))=int_(0)^(pi//2)(sec^(2)theta d theta)/((tan theta+sec theta)^(3))`
`=int_(0)^(pi//2)(1//cos^(2) theta)/(((sin theta)/(cos theta)+(1)/(cos theta))^(3))d theta=int_(0)^(pi//2)(cos theta)/(( sin theta+1)^(3))d theta`
`=[-(1)/(2(sin theta+1)^(2))]_(0)^(pi//2)=(-1)/(8)+(1)/(2)=(3)/(8)`