integral of cos^3(X)sin^2(X)DX

解

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

D (\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 cos^{3} (X) sin^{2} (X) D X d X

定数を除く:

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

= D \cdot \int cos^{3} (X) sin^{2} (X) X d X

部分積分を適用する

= D (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)

= D (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

D (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))) : D (\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))

= D (\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

= D (\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