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int(sqrt(x^2+1))/(x^4)dx=...

`int(sqrt(x^2+1))/(x^4)dx=`

A

`-(1)/(3)((x^(2)+1)^(3//2))/(x^(3))+C`

B

`x^(3)(x^(2)+1)^(-1//2)+C`

C

`(sqrt(x^(2)+1))/(x^(2))+C`

D

`-(1)/(3)((x^(2)+1)^(3//2))/(x^(2))+C`

Text Solution

Verified by Experts

The correct Answer is:
A
Put `x=tan theta " so that " sqrt(x^(2)+1)=sec theta, dx=sec^(2) theta d theta`
` :. I=int(sec theta sec^(2) theta)/(tan^(4)theta)d theta=int (cos theta)/(sin^(4)theta)d theta`
`= -(1)/(3)(1)/(sin^(3)theta)+C`
`= -(1)/(3)(sec^(2)theta)/(tan^(3)theta)+C`
`= -(1)/(3)((x^(2)+1)^(3//2))/(x^(3))+C`
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