integral of sin(x)cos(x)ln(sin(x))

解

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

\frac{1}{2} sin^{2} (x) ln (sin (x)) - \frac{sin ^{2} ( x )}{4} + C

解答ステップ

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

u置換積分法を適用する

= \int u ln (u) d u

部分積分を適用する

= \frac{1}{2} u^{2} ln (u) - \int \frac{u}{2} d u

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

= \frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4}

代用を戻す

u = sin (x)

= \frac{1}{2} sin^{2} (x) ln (sin (x)) - \frac{sin ^{2} ( x )}{4}

定数を解答に追加する

= \frac{1}{2} sin^{2} (x) ln (sin (x)) - \frac{sin ^{2} ( x )}{4} + C