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

解

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

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

解答ステップ

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

部分積分を適用する

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

\int \frac{sin ^{2} ( x )}{2 x} d x = \frac{1}{4} (ln ∣ x ∣ - Ci (2 x))

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

定数を解答に追加する

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