What is the integral of sin(2x)ln(tan(x)) ?

The integral of sin(2x)ln(tan(x)) is 1/2 ln|sin(2x)|-1/2 cos(2x)ln(tan(x))+C

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integral of sin(2x)ln(tan(x))

\int sin (2 x) ln (tan (x)) d x

\frac{1}{2} ln ∣ sin (2 x) ∣ - \frac{1}{2} cos (2 x) ln (tan (x)) + C

Solution steps

\int sin (2 x) ln (tan (x)) d x

Apply Integration By Parts

: - \int - cot (2 x) d x - \frac{1}{2} cos (2 x) ln (tan (x))

= - \int - cot (2 x) d x - \frac{1}{2} cos (2 x) ln (tan (x))

\int - cot (2 x) d x = - \frac{1}{2} ln ∣ sin (2 x) ∣

= - (- \frac{1}{2} ln ∣ sin (2 x) ∣) - \frac{1}{2} cos (2 x) ln (tan (x))

Simplify

= \frac{1}{2} ln ∣ sin (2 x) ∣ - \frac{1}{2} cos (2 x) ln (tan (x))

Add a constant to the solution

= \frac{1}{2} ln ∣ sin (2 x) ∣ - \frac{1}{2} cos (2 x) ln (tan (x)) + C

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What is the integral of sin(2x)ln(tan(x)) ?
The integral of sin(2x)ln(tan(x)) is 1/2 ln|sin(2x)|-1/2 cos(2x)ln(tan(x))+C