integral of (4x^2)/(\sqrt[3]{1-x)}

解

\int \frac{4 x ^{2}}{3 1 - x} d x

4 (- \frac{3}{2} x^{2} (1 - x)^{\frac{2}{3}} - 3 (- \frac{3}{8} (1 - x)^{\frac{8}{3}} + \frac{3}{5} (1 - x)^{\frac{5}{3}})) + C

解答ステップ

\int \frac{4 x ^{2}}{3 1 - x} d x

定数を除く:

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

= 4 \cdot \int \frac{x ^{2}}{3 1 - x} d x

部分積分を適用する

= 4 (- \frac{3}{2} x^{2} (1 - x)^{\frac{2}{3}} - \int - 3 x (1 - x)^{\frac{2}{3}} d x)

\int - 3 x (1 - x)^{\frac{2}{3}} d x = 3 (- \frac{3}{8} (1 - x)^{\frac{8}{3}} + \frac{3}{5} (1 - x)^{\frac{5}{3}})

= 4 (- \frac{3}{2} x^{2} (1 - x)^{\frac{2}{3}} - 3 (- \frac{3}{8} (1 - x)^{\frac{8}{3}} + \frac{3}{5} (1 - x)^{\frac{5}{3}}))

定数を解答に追加する

= 4 (- \frac{3}{2} x^{2} (1 - x)^{\frac{2}{3}} - 3 (- \frac{3}{8} (1 - x)^{\frac{8}{3}} + \frac{3}{5} (1 - x)^{\frac{5}{3}})) + C