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  4. If ∫ cos x log(tan (x/2))dx = sin x log(tan (x/2))+ f(x) then f(x) is equal to, (assuming c is a arbitrary real constant)

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Q. If $\int\,cos\,x\,log\left(tan \frac{x}{2}\right)dx = sin\,x\,log\left(tan \frac{x}{2}\right)+$ f(x) then f(x) is equal to, (assuming c is a arbitrary real constant)

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Solution:

$IBP \Rightarrow I = sinx\,log\left(tan \frac{x}{2}\right)-x + c$
$\therefore \,f\left(x\right) = c - x$