integral of x^2sin(4x)

解

\int x^{2} sin (4 x) d x

- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) + C

解答ステップ

\int x^{2} sin (4 x) d x

部分積分を適用する

= - \frac{1}{4} x^{2} cos (4 x) - \int - \frac{1}{2} x cos (4 x) d x

\int - \frac{1}{2} x cos (4 x) d x = - \frac{1}{32} (4 x sin (4 x) + cos (4 x))

= - \frac{1}{4} x^{2} cos (4 x) - (- \frac{1}{32} (4 x sin (4 x) + cos (4 x)))

簡素化

= - \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))

定数を解答に追加する

= - \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) + C