Integrate the following w.r.t.x: `(1)/(x^(2)sqrt(a^(2)+x^(2)))`

Explanation

`"Let I"=int (1)/(x^(2)sqrt(a^(2)+x^(2)))dx`
Put x=a tan `theta`
`therefore dx=a sec^(2)theta d theta and tantheta=(x)/(a)`
`therefore I=int (1)/(a^(2)tan^(2)thetasqrt(a^(2)+a^(2)tan theta)).a sec^(2)d theta`
`=int (1)/(a^(2)tan^(2)theta.asec theta).asec^(2)thetad theta`
`=(1)/(a^(2))int (sec theta)/(tan^(2)theta)d theta=(1)/(a^(2))iint(1)/(cos theta)xx(cos^(2)theta)/(sin^(2)theta)d theta`
`=(1)/(a^(2))int(cos theta)/(sin^(2)theta)d theta=(1)/(a^(2)) int (1)/(sintheta).(cos theta)/(sin theta)d theta`
`=(1)/(a^(2))int cosec theta.cot theta d theta`
`=-(1)/(a^(2))(-cosectheta)+c`
`=-(1)/(a^(2)).cosectheta+c`...(1)
`Now, tan theta=(x)/(a)" "cot theta=(a)/(x)`
`therefore cosectheta=sqrt(1+cot^(2)theta)=sqrt(1+(a^(2))/(x^(2)))`
`=sqrt(x^(2)+a^(2))/(x)`
`therefore "From"(1),I=-(1)/(a^(2)).sqrt(x^(2)+a^(2))/(x)+c`
`=-(sqrt(a^(2)+x^(2)))/(a^(2).x)+c`