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人気のある微分積分 >

integral of x^3sqrt(5+x^2)

解

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

解

\frac{1}{5} (5 + x^{2})^{\frac{5}{2}} - \frac{5}{3} (5 + x^{2})^{\frac{3}{2}} + C

解答ステップ

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

u置換積分法を適用する

= \int (u^{2} - 5) u^{2} d u

拡張

(u^{2} - 5) u^{2} : u^{4} - 5 u^{2}

= \int u^{4} - 5 u^{2} 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^{4} d u - \int 5 u^{2} d u

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

\int 5 u^{2} d u = \frac{5 u ^{3}}{3}

= \frac{u ^{5}}{5} - \frac{5 u ^{3}}{3}

代用を戻す

u = 5 + x^{2}

= \frac{( 5 + x ^{2} ) ^{5}}{5} - \frac{5 ( 5 + x ^{2} ) ^{3}}{3}

簡素化

\frac{( 5 + x ^{2} ) ^{5}}{5} - \frac{5 ( 5 + x ^{2} ) ^{3}}{3} : \frac{1}{5} (5 + x^{2})^{\frac{5}{2}} - \frac{5}{3} (5 + x^{2})^{\frac{3}{2}}

= \frac{1}{5} (5 + x^{2})^{\frac{5}{2}} - \frac{5}{3} (5 + x^{2})^{\frac{3}{2}}

定数を解答に追加する

= \frac{1}{5} (5 + x^{2})^{\frac{5}{2}} - \frac{5}{3} (5 + x^{2})^{\frac{3}{2}} + C

グラフ

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