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Evaluate int(dx)/(sqrt(x^2+2x-3))...

Evaluate `int(dx)/(sqrt(x^2+2x-3))`

Text Solution

Verified by Experts

`I=int (dx)/((x+1)^(3)sqrt((x+1)^(2)-4))`
Let `x+1=2 sec theta`
` :. dx=2 sec theta * tan theta * d theta`
` :. I=int(2sec theta .tan theta d theta)/(sec^(3)theta 2 tan theta)`
`=cos^(2)theta d theta`
`=int(1+cos 2theta)/(2) d theta`
`=(1)/(2)(theta + (sin 2theta)/(2))+c`
`=(1)/(2)"sec"^(-1)(x+1)/(2)+(1)/(2)sin theta .cos theta +c`
`=(1)/(2)"sec"^(-1)(x+1)/(2)+(1)/(2)sqrt(1-(4)/((x+1)^(2))).(2)/((x+1))+c`
`=(1)/(2)"sec"^(-1)(x+1)/(2)+(1)/((x+1)^(2))sqrt(x^(2)+2x-3)+c`
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