integral of (x^3-5x^2+2x+4)sin(4x)

解答

\int (x^{3} - 5 x^{2} + 2 x + 4) sin (4 x) d x

- \frac{1}{4} x^{3} cos (4 x) + \frac{5}{4} x^{2} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) - \frac{5}{8} x sin (4 x) - \frac{13}{32} x cos (4 x) + \frac{13}{128} sin (4 x) - \frac{37}{32} cos (4 x) + C

求解步骤

\int (x^{3} - 5 x^{2} + 2 x + 4) sin (4 x) d x

乘开

(x^{3} - 5 x^{2} + 2 x + 4) sin (4 x) : x^{3} sin (4 x) - 5 x^{2} sin (4 x) + 2 x sin (4 x) + 4 sin (4 x)

= \int x^{3} sin (4 x) - 5 x^{2} sin (4 x) + 2 x sin (4 x) + 4 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^{3} sin (4 x) d x - \int 5 x^{2} sin (4 x) d x + \int 2 x sin (4 x) d x + \int 4 sin (4 x) d x

\int x^{3} sin (4 x) d x = - \frac{1}{4} x^{3} cos (4 x) + \frac{3}{128} (8 x^{2} sin (4 x) + 4 x cos (4 x) - sin (4 x))

\int 5 x^{2} sin (4 x) d x = 5 (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)))

\int 2 x sin (4 x) d x = 2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

\int 4 sin (4 x) d x = - cos (4 x)

= - \frac{1}{4} x^{3} cos (4 x) + \frac{3}{128} (8 x^{2} sin (4 x) + 4 x cos (4 x) - sin (4 x)) - 5 (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))) + 2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) - cos (4 x)

化简

- \frac{1}{4} x^{3} cos (4 x) + \frac{3}{128} (8 x^{2} sin (4 x) + 4 x cos (4 x) - sin (4 x)) - 5 (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))) + 2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) - cos (4 x) : - \frac{1}{4} x^{3} cos (4 x) + \frac{5}{4} x^{2} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) - \frac{5}{8} x sin (4 x) - \frac{13}{32} x cos (4 x) + \frac{13}{128} sin (4 x) - \frac{37}{32} cos (4 x)

= - \frac{1}{4} x^{3} cos (4 x) + \frac{5}{4} x^{2} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) - \frac{5}{8} x sin (4 x) - \frac{13}{32} x cos (4 x) + \frac{13}{128} sin (4 x) - \frac{37}{32} cos (4 x)

解答补常数

= - \frac{1}{4} x^{3} cos (4 x) + \frac{5}{4} x^{2} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) - \frac{5}{8} x sin (4 x) - \frac{13}{32} x cos (4 x) + \frac{13}{128} sin (4 x) - \frac{37}{32} cos (4 x) + C