integral of x^3(x^2-1)^{3/2}

解

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

\frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} - \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}} + C

解答ステップ

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

部分積分を適用する

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \int \frac{3}{4} x^{5} (x^{2} - 1)^{\frac{1}{2}} d x

\int \frac{3}{4} x^{5} (x^{2} - 1)^{\frac{1}{2}} d x = \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} + \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} + \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}}

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - (\frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} + \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} + \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}})

簡素化

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} - \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}}

定数を解答に追加する

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} - \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}} + C