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人気のある微分積分 >

integral of x/(\sqrt[4]{x+9)}

解

\int \frac{x}{4 x + 9} d x

解

\frac{4}{7} (x + 9)^{\frac{7}{4}} - 12 (x + 9)^{\frac{3}{4}} + C

解答ステップ

\int \frac{x}{4 x + 9} d x

u置換積分法を適用する

= \int \frac{u - 9}{4 u} d u

拡張

\frac{u - 9}{4 u} : u^{\frac{3}{4}} - \frac{9}{4 u}

= \int u^{\frac{3}{4}} - \frac{9}{4 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

= \int u^{\frac{3}{4}} d u - \int \frac{9}{4 u} d u

\int u^{\frac{3}{4}} d u = \frac{4}{7} u^{\frac{7}{4}}

\int \frac{9}{4 u} d u = 12 u^{\frac{3}{4}}

= \frac{4}{7} u^{\frac{7}{4}} - 12 u^{\frac{3}{4}}

代用を戻す

u = x + 9

= \frac{4}{7} (x + 9)^{\frac{7}{4}} - 12 (x + 9)^{\frac{3}{4}}

定数を解答に追加する

= \frac{4}{7} (x + 9)^{\frac{7}{4}} - 12 (x + 9)^{\frac{3}{4}} + C

グラフ

人気の例

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\frac{d}{d x} (x^{3} + 2 x^{2} - x - 1)

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