integral of-sin(ln(x))ln(x)

解

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

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

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int sin (ln (x)) ln (x) d x

部分積分を適用する

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

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

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

簡素化

- (ln (x) (\frac{1}{2} x sin (ln (x)) - \frac{1}{2} x cos (ln (x))) - (- \frac{1}{2} x cos (ln (x)))) : - \frac{1}{2} x ln (x) sin (ln (x)) + \frac{1}{2} x ln (x) cos (ln (x)) - \frac{1}{2} x cos (ln (x))

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

定数を解答に追加する

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