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

解答

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

- \frac{1}{4} x^{3} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) + \frac{3}{32} x cos (4 x) - \frac{3}{128} sin (4 x) + x^{2} cos (4 x) - \frac{1}{2} x sin (4 x) - \frac{11}{8} cos (4 x) + C

求解步骤

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

乘开

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

= \int x^{3} sin (4 x) - 4 x^{2} sin (4 x) + 5 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 4 x^{2} sin (4 x) d x + \int 5 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 4 x^{2} sin (4 x) d x = - x^{2} cos (4 x) + \frac{1}{2} x sin (4 x) + \frac{1}{8} cos (4 x)

\int 5 sin (4 x) d x = - \frac{5}{4} 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)) - (- x^{2} cos (4 x) + \frac{1}{2} x sin (4 x) + \frac{1}{8} cos (4 x)) - \frac{5}{4} 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)) - (- x^{2} cos (4 x) + \frac{1}{2} x sin (4 x) + \frac{1}{8} cos (4 x)) - \frac{5}{4} cos (4 x) : - \frac{1}{4} x^{3} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) + \frac{3}{32} x cos (4 x) - \frac{3}{128} sin (4 x) + x^{2} cos (4 x) - \frac{1}{2} x sin (4 x) - \frac{11}{8} cos (4 x)

= - \frac{1}{4} x^{3} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) + \frac{3}{32} x cos (4 x) - \frac{3}{128} sin (4 x) + x^{2} cos (4 x) - \frac{1}{2} x sin (4 x) - \frac{11}{8} cos (4 x)

解答补常数

= - \frac{1}{4} x^{3} cos (4 x) + \frac{3}{16} x^{2} sin (4 x) + \frac{3}{32} x cos (4 x) - \frac{3}{128} sin (4 x) + x^{2} cos (4 x) - \frac{1}{2} x sin (4 x) - \frac{11}{8} cos (4 x) + C

\int e^{x (a - s)} d x

\int \frac{5}{x ^{2}} d x

\int \frac{1}{4 x ^{2} - 16} d x

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

\int 9 - X^{2} d X