Question

Show that:  `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

Answer 1

`Let I =int1/(x^2sqrt(a^2+x^2)dx`

`Put x = a tantheta`

Differentiating w.r.t. theta we get

`dx = a sec^2 theta d theta`

`theta=tan^-1(x/a)`

`I=int(asec^2theta d theta)/(a^2tan^2thetasqrt(a^2+a^2tan^2theta))`

`=1/a^2intsectheta/tan^2theta d theta`

`=1/a^2intcostheta/sin^2thetad theta`

`=1/a^2intcosecthetacotthetad theta`

`I=-1/a^2cosectheta+c ....(i)`

`But tantheta=x/a`

`cottheta`=a/x`

`cosec^2theta`=1+cot^2theta`

`cosec^2theta=1+a^2/x^2`

`cosec^2theta=(x^2+a^2)/x^2`

`cosectheta=sqrt(x^2_a^2)/x.........(ii)`

`I=-1/a^2sqrt(x^2+a^2)/x+c  `