integral of x^2sin^2(x/2)

解

\int x^{2} sin^{2} (\frac{x}{2}) d x

\frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))) + C

解答ステップ

\int x^{2} sin^{2} (\frac{x}{2}) d x

部分積分を適用する

= \frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \int \frac{1}{6} x^{3} sin (x) d x

\int \frac{1}{6} x^{3} sin (x) d x = \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))

= \frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))

定数を解答に追加する

= \frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))) + C