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人気のある微分積分 >

integral of xarcsin(2x)

解

\int x arcsin (2 x) d x

解

\frac{1}{4} (2 x^{2} arcsin (2 x) - \frac{1}{4} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x)))) + C

解答ステップ

\int x arcsin (2 x) d x

u置換積分法を適用する

= \int \frac{u arcsin ( u )}{4} d u

定数を除く:

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

= \frac{1}{4} \cdot \int u arcsin (u) d u

部分積分を適用する

= \frac{1}{4} (\frac{1}{2} u^{2} arcsin (u) - \int \frac{u ^{2}}{2 1 - u ^{2}} d u)

\int \frac{u ^{2}}{2 1 - u ^{2}} d u = \frac{1}{4} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u)))

= \frac{1}{4} (\frac{1}{2} u^{2} arcsin (u) - \frac{1}{4} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))))

代用を戻す

u = 2 x

= \frac{1}{4} (\frac{1}{2} (2 x)^{2} arcsin (2 x) - \frac{1}{4} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x))))

\frac{1}{2} (2 x)^{2} arcsin (2 x) = 2 x^{2} arcsin (2 x)

= \frac{1}{4} (2 x^{2} arcsin (2 x) - \frac{1}{4} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x))))

定数を解答に追加する

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

グラフ

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