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