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

解

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

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

解答ステップ

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

部分積分を適用する

= - \frac{1}{2} ln (cos (x)) cos (2 x) - \int \frac{1}{2} tan (x) cos (2 x) d x

\int \frac{1}{2} tan (x) cos (2 x) d x = \frac{1}{2} (\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x)))

= - \frac{1}{2} ln (cos (x)) cos (2 x) - \frac{1}{2} (\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x)))

定数を解答に追加する

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