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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int sin (x) ln (sin (x)) d x

Apply Integration By Parts

: - ln (sin (x)) cos (x) - \int - cot (x) cos (x) d x

= 2 (- ln (sin (x)) cos (x) - \int - cot (x) cos (x) d x)

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

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

Simplify

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

Add a constant to the solution

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

\int x + \frac{2}{x} d x

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