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Evaluate : int(log(e)(1+sin^(2)x))/(cos^...

Evaluate : `int(log_(e)(1+sin^(2)x))/(cos^(2)x)dx`

Text Solution

Verified by Experts

The correct Answer is:
`log_(e)(1+sin^(2)x).tanx-2x+sqrt2tan^(-1)(sqrt2tanx)+C`
`int(log_(e)(1+sin^(2)x)dx)/(cos^(2)x)`
`=log_(e)(1+sin^(2)x).tanx-2int(sin^(2)x+1-1)/(1+sin^(2)x)`
`=log_(e)(1+sin^(2)x).tanx-2x+2 int(sec^(2)xdx)/(1+2tan^(2)x)`
`=log_(e)(1+sin^(2)x).tanx-2x+sqrt2tan^(-1)(sqrt2tanx)+C`
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