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int{(logx-1)/(1+(logx)^(2))}^(2) dx is e...

`int{(logx-1)/(1+(logx)^(2))}^(2)` dx is equal to

A

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

B

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

C

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

D

`(logx)/((logx)^(2)+1)+C`

Text Solution

Verified by Experts

The correct Answer is:
A
Let us assume a function `f(x)=(x)/((logx)^(2)+1)+C` from option (a).
`therefore" "(d)/(dx)((x)/((logx)^(2)+1)+C)`
`=((logx)^(2)+1-x(2logx.(1)/(x)))/([(logx)^(2)+1]^(2))`
`=((logx)^(2)+1-2logx)/([(logx)^(2)+1]^(2))`
`=((logx-1)^(2))/([(logx)^(2)+1]^(2))=[(logx-1)/(1+(logx)^(2))]^(2)`
Hence, this result is equal to the given integral.
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