integral of xsin^2(x)cos^3(x)

解

\int x sin^{2} (x) cos^{3} (x) d x

\frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x) + C

解答ステップ

\int x sin^{2} (x) cos^{3} (x) d x

部分積分を適用する

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

\int \frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x) d x = - \frac{2}{15} cos (x) + \frac{1}{9} cos^{3} (x) - \frac{2}{15} cos^{3} (x) + \frac{1}{25} cos^{5} (x)

= x (\frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x)) - (- \frac{2}{15} cos (x) + \frac{1}{9} cos^{3} (x) - \frac{2}{15} cos^{3} (x) + \frac{1}{25} cos^{5} (x))

簡素化

x (\frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x)) - (- \frac{2}{15} cos (x) + \frac{1}{9} cos^{3} (x) - \frac{2}{15} cos^{3} (x) + \frac{1}{25} cos^{5} (x)) : \frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x)

= \frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x)

定数を解答に追加する

= \frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x) + C