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

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

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

\int \frac{ln ( sin ( x ) )}{( sin ( x ) ) ^{2}} d x

- ln (sin (x)) cot (x) - x - cot (x) + C

Solution steps

\int \frac{ln ( sin ( x ) )}{( sin ( x ) ) ^{2}} d x

Factor

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

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

Apply Integration By Parts

: - ln (sin (x)) cot (x) - \int - cot^{2} (x) d x

= - ln (sin (x)) cot (x) - \int - cot^{2} (x) d x

\int - cot^{2} (x) d x = x + cot (x)

= - ln (sin (x)) cot (x) - (x + cot (x))

Simplify

= - ln (sin (x)) cot (x) - x - cot (x)

Add a constant to the solution

= - ln (sin (x)) cot (x) - x - cot (x) + C

\int 4 x^{2} ln (\frac{x}{2}) d x

\int csc^{2} (x) sin (x) d x

\int \frac{2 x ^{3} + 3 x}{x ^{4} + 3 x ^{2} - 2} d x

\int \frac{x - 1}{x + 3} d x

\int \frac{2}{x ^{3} - x ^{2} + x - 1} d x

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