What is the integral of sin(x)ln(tan(x)) ?

The integral of sin(x)ln(tan(x)) is -ln(tan(x))cos(x)+ln|tan(x/2)|+C

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integral of sin(x)ln(tan(x))

\int sin (x) ln (tan (x)) d x

- ln (tan (x)) cos (x) + ln tan (\frac{x}{2}) + C

Solution steps

\int sin (x) ln (tan (x)) d x

Apply Integration By Parts

: - ln (tan (x)) cos (x) - \int - 2 csc (2 x) cos (x) d x

= - ln (tan (x)) cos (x) - \int - 2 csc (2 x) cos (x) d x

\int - 2 csc (2 x) cos (x) d x = - ln tan (\frac{x}{2})

= - ln (tan (x)) cos (x) - (- ln tan (\frac{x}{2}))

Simplify

= - ln (tan (x)) cos (x) + ln tan (\frac{x}{2})

Add a constant to the solution

= - ln (tan (x)) cos (x) + ln tan (\frac{x}{2}) + C

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What is the integral of sin(x)ln(tan(x)) ?
The integral of sin(x)ln(tan(x)) is -ln(tan(x))cos(x)+ln|tan(x/2)|+C