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

解

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

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

簡素化

= \int x^{2} cos^{2} (\frac{x}{2}) d x

部分積分を適用する

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

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

簡素化

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