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人気のある微分積分 >

integral of (x^3)/(\sqrt[5]{x^2+9)}

解

\int \frac{x ^{3}}{5 x ^{2} + 9} d x

解

\frac{1}{2} (\frac{5}{9} (x^{2} + 9)^{\frac{9}{5}} - \frac{45}{4} (x^{2} + 9)^{\frac{4}{5}}) + C

解答ステップ

\int \frac{x ^{3}}{5 x ^{2} + 9} d x

u置換積分法を適用する

= \int \frac{u - 9}{2 5 u} d u

定数を除く:

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

= \frac{1}{2} \cdot \int \frac{u - 9}{5 u} d u

拡張

\frac{u - 9}{5 u} : u^{\frac{4}{5}} - \frac{9}{5 u}

= \frac{1}{2} \cdot \int u^{\frac{4}{5}} - \frac{9}{5 u} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int u^{\frac{4}{5}} d u - \int \frac{9}{5 u} d u)

\int u^{\frac{4}{5}} d u = \frac{5}{9} u^{\frac{9}{5}}

\int \frac{9}{5 u} d u = \frac{45}{4} u^{\frac{4}{5}}

= \frac{1}{2} (\frac{5}{9} u^{\frac{9}{5}} - \frac{45}{4} u^{\frac{4}{5}})

代用を戻す

u = x^{2} + 9

= \frac{1}{2} (\frac{5}{9} (x^{2} + 9)^{\frac{9}{5}} - \frac{45}{4} (x^{2} + 9)^{\frac{4}{5}})

定数を解答に追加する

= \frac{1}{2} (\frac{5}{9} (x^{2} + 9)^{\frac{9}{5}} - \frac{45}{4} (x^{2} + 9)^{\frac{4}{5}}) + C

グラフ

人気の例

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t^{2} y^{^{''}} - 3 t y^{^{'}} + 3 y = 0

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\int \frac{1}{y ^{2} ( y ^{2} + 1 )} d y

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limit as x approaches-infinity of 4^x

x \to - \infty lim (4^{x})