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

The integral of ln(sin(x))cot(x) is (ln^2(sin(x)))/2+C

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integral of ln(sin(x))cot(x)

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

\frac{ln ^{2} ( sin ( x ) )}{2} + C

Solution steps

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

Rewrite using trig identities

= \int cos (x) ln (sin (x)) csc (x) d x

Apply u-substitution

: \int \frac{ln ( u )}{u} d u

= \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{v ^{2}}{2}

Substitute back

= \frac{ln ^{2} ( sin ( x ) )}{2}

Add a constant to the solution

= \frac{ln ^{2} ( sin ( x ) )}{2} + C

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What is the integral of ln(sin(x))cot(x) ?
The integral of ln(sin(x))cot(x) is (ln^2(sin(x)))/2+C