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

解

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

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

解答ステップ

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

因数

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

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

部分積分を適用する

= - 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))

簡素化

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

定数を解答に追加する

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