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人気のある微分積分 >

integral of 4x^4(27+x^5)^{1/3}

解

\int 4 x^{4} (27 + x^{5})^{\frac{1}{3}} d x

解

\frac{3}{5} (27 + x^{5})^{\frac{4}{3}} + C

解答ステップ

\int 4 x^{4} (27 + x^{5})^{\frac{1}{3}} d x

定数を除く:

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

= 4 \cdot \int x^{4} (27 + x^{5})^{\frac{1}{3}} d x

u置換積分法を適用する

= 4 \cdot \int \frac{u ^{\frac{1}{3}}}{5} d u

定数を除く:

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

= 4 \cdot \frac{1}{5} \cdot \int u^{\frac{1}{3}} d u

べき乗則を適用する

= 4 \cdot \frac{1}{5} \cdot \frac{3}{4} u^{\frac{4}{3}}

代用を戻す

u = 27 + x^{5}

= 4 \cdot \frac{1}{5} \cdot \frac{3}{4} (27 + x^{5})^{\frac{4}{3}}

簡素化

4 \cdot \frac{1}{5} \cdot \frac{3}{4} (27 + x^{5})^{\frac{4}{3}} : \frac{3}{5} (27 + x^{5})^{\frac{4}{3}}

= \frac{3}{5} (27 + x^{5})^{\frac{4}{3}}

定数を解答に追加する

= \frac{3}{5} (27 + x^{5})^{\frac{4}{3}} + C

グラフ

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