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人気のある微分積分 >

integral from 0 to 1/3 of xarcsin(3x)

解

\int_{0}^{\frac{1}{3}} x arcsin (3 x) d x

解

\frac{π}{72}

+1

十進法表記

0.04363 \dots

解答ステップ

\int_{0}^{\frac{1}{3}} x arcsin (3 x) d x

u置換積分法を適用する

= \int_{0}^{1} \frac{u arcsin ( u )}{9} d u

定数を除く:

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

= \frac{1}{9} \cdot \int_{0}^{1} u arcsin (u) d u

部分積分を適用する

= \frac{1}{9} [\frac{1}{2} u^{2} arcsin (u) - \int \frac{u ^{2}}{2 1 - u ^{2}} d u]_{0}^{1}

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

= \frac{1}{9} [\frac{1}{2} u^{2} arcsin (u) - \frac{1}{4} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u)))]_{0}^{1}

限度を計算する:

\frac{π}{8}

= \frac{1}{9} \cdot \frac{π}{8}

簡素化

= \frac{π}{72}

グラフ

人気の例

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x \to - 6 lim (\frac{x ^{2} - 9}{6 - x})

integral of (x+4)^9

\int (x + 4)^{9} d x

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