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人気のある微分積分 >

integral of sin(ln(7x))

解

\int sin (ln (7 x)) d x

解

\frac{1}{2} x (sin (ln (7 x)) - cos (ln (7 x))) + C

解答ステップ

\int sin (ln (7 x)) d x

u置換積分法を適用する

= \int sin (ln (u)) \frac{1}{7} d u

定数を除く:

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

= \frac{1}{7} \cdot \int sin (ln (u)) d u

部分積分を適用する

= \frac{1}{7} (u sin (ln (u)) - \int cos (ln (u)) d u)

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

= \frac{1}{7} (u sin (ln (u)) - (\frac{1}{2} u sin (ln (u)) + \frac{1}{2} u cos (ln (u))))

代用を戻す

u = 7 x

= \frac{1}{7} (7 x sin (ln (7 x)) - (\frac{1}{2} \cdot 7 x sin (ln (7 x)) + \frac{1}{2} \cdot 7 x cos (ln (7 x))))

簡素化

\frac{1}{7} (7 x sin (ln (7 x)) - (\frac{1}{2} \cdot 7 x sin (ln (7 x)) + \frac{1}{2} \cdot 7 x cos (ln (7 x)))) : \frac{1}{2} x (sin (ln (7 x)) - cos (ln (7 x)))

= \frac{1}{2} x (sin (ln (7 x)) - cos (ln (7 x)))

定数を解答に追加する

= \frac{1}{2} x (sin (ln (7 x)) - cos (ln (7 x))) + C

グラフ

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