integral of 3x^3cos(5x)

解

\int 3 x^{3} cos (5 x) d x

3 (\frac{1}{5} x^{3} sin (5 x) - \frac{3}{5} (- \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x)))) + C

解答ステップ

\int 3 x^{3} cos (5 x) d x

定数を除く:

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

= 3 \cdot \int x^{3} cos (5 x) d x

部分積分を適用する

= 3 (\frac{1}{5} x^{3} sin (5 x) - \int \frac{3}{5} x^{2} sin (5 x) d x)

\int \frac{3}{5} x^{2} sin (5 x) d x = \frac{3}{5} (- \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x)))

= 3 (\frac{1}{5} x^{3} sin (5 x) - \frac{3}{5} (- \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x))))

定数を解答に追加する

= 3 (\frac{1}{5} x^{3} sin (5 x) - \frac{3}{5} (- \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x)))) + C