1.

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

Answer» `I=int(1)/(x^(2)sqrt(a^(2)+x^(2)))dx`
Let `x=atantheta`
`dx=asec^(2)thetad theta`
`I=int(asec^(2)thetad theta)/(a^(2)tan^(2)thetasqrt(a^(2)+a^(2)tan^(2)theta))`
`I=int(sec^(2)theta)/(a^(2)tan^(2)thetasqrt(1+tan^(2)theta))d theta`
`=(1)/(a^(2))int(sec^(2)theta)/(tan^(2)thetasectheta)d theta`
`=(1)/(a^(2))int(1xxcos^(2)theta)/(costhetaxxsin^(2)theta)d theta`
`=(1)/(a^(2))int(costheta)/(sin^(2)theta)d theta`
Put `sintheta=t`
`rArrcosthetad theta=dt`
`I=(1)/(a^(2))int(dt)/(t^(2))`
`=(-1)/(a^(2)t)+c`
`=-(1)/(a^(2)sintheta)+c`
`=-(cosectheta)/(a^(2))+c`
`=-(sqrt(1+cot^(2)theta))/(a^(2))+c`
`=-(sqrt(1+(a^(2))/(x^(2))))/(a^(2))+c`
`=-(sqrt(x^(2)+a^(2)))/(a^(2)x)+c`


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