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

解

\int (5 - x^{2})^{3} d x

x (5 - x^{2})^{\frac{3}{2}} + \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x))) + C

解答ステップ

\int (5 - x^{2})^{3} d x

部分積分を適用する

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

\int - 3 x^{2} 5 - x^{2} d x = - \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x)))

= x (5 - x^{2})^{3} - (- \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x))))

簡素化

x (5 - x^{2})^{3} - (- \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x)))) : x (5 - x^{2})^{\frac{3}{2}} + \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x)))

= x (5 - x^{2})^{\frac{3}{2}} + \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x)))

定数を解答に追加する

= x (5 - x^{2})^{\frac{3}{2}} + \frac{75}{8} (arcsin (\frac{1}{5} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{5} x))) + C