integral of 3sin(ln(x))

解

\int 3 sin (ln (x)) d x

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

解答ステップ

\int 3 sin (ln (x)) d x

定数を除く:

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

= 3 \cdot \int sin (ln (x)) d x

部分積分を適用する

= 3 (x sin (ln (x)) - \int cos (ln (x)) d x)

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

= 3 (x sin (ln (x)) - (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x))))

拡張

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

= 3 (\frac{1}{2} x sin (ln (x)) - \frac{1}{2} x cos (ln (x)))

定数を解答に追加する

= 3 (\frac{1}{2} x sin (ln (x)) - \frac{1}{2} x cos (ln (x))) + C