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人気のある微分積分 >

integral of sin(ln(3x))

解

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

解

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

解答ステップ

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

u置換積分法を適用する

= \int sin (ln (u)) \frac{1}{3} d u

定数を除く:

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

= \frac{1}{3} \cdot \int sin (ln (u)) d u

部分積分を適用する

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

代用を戻す

u = 3 x

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

derivative of 4/(x^2+2)

\frac{d}{d x} (\frac{4}{x ^{2} + 2})

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\int_{0}^{l n (2)} e^{2 x} d x

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\int - \frac{y}{3 + y ^{2}} d y