integral of (x^2-8x)sin(4x)

解答

\int (x^{2} - 8 x) sin (4 x) d x

- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) - 8 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

求解步骤

\int (x^{2} - 8 x) sin (4 x) d x

乘开

(x^{2} - 8 x) sin (4 x) : x^{2} sin (4 x) - 8 x sin (4 x)

= \int x^{2} sin (4 x) - 8 x sin (4 x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} sin (4 x) d x - \int 8 x sin (4 x) d x

\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))

\int 8 x sin (4 x) d x = 8 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

= - \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) - 8 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

解答补常数

= - \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) - 8 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

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