integral of 5x^2sin(9x)

解

\int 5 x^{2} sin (9 x) d x

5 (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x))) + C

解答ステップ

\int 5 x^{2} sin (9 x) d x

定数を除く:

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

= 5 \cdot \int x^{2} sin (9 x) d x

部分積分を適用する

= 5 (- \frac{1}{9} x^{2} cos (9 x) - \int - \frac{2}{9} x cos (9 x) d x)

\int - \frac{2}{9} x cos (9 x) d x = - \frac{2}{729} (9 x sin (9 x) + cos (9 x))

= 5 (- \frac{1}{9} x^{2} cos (9 x) - (- \frac{2}{729} (9 x sin (9 x) + cos (9 x))))

簡素化

= 5 (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x)))

定数を解答に追加する

= 5 (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x))) + C