integral of x^3cos(12x)

解

\int x^{3} cos (12 x) d x

\frac{1}{12} x^{3} sin (12 x) - \frac{1}{4} (- \frac{1}{12} x^{2} cos (12 x) + \frac{1}{864} (12 x sin (12 x) + cos (12 x))) + C

解答ステップ

\int x^{3} cos (12 x) d x

部分積分を適用する

= \frac{1}{12} x^{3} sin (12 x) - \int \frac{1}{4} x^{2} sin (12 x) d x

\int \frac{1}{4} x^{2} sin (12 x) d x = \frac{1}{4} (- \frac{1}{12} x^{2} cos (12 x) + \frac{1}{864} (12 x sin (12 x) + cos (12 x)))

= \frac{1}{12} x^{3} sin (12 x) - \frac{1}{4} (- \frac{1}{12} x^{2} cos (12 x) + \frac{1}{864} (12 x sin (12 x) + cos (12 x)))

定数を解答に追加する

= \frac{1}{12} x^{3} sin (12 x) - \frac{1}{4} (- \frac{1}{12} x^{2} cos (12 x) + \frac{1}{864} (12 x sin (12 x) + cos (12 x))) + C