integral of cos(4x)cos(x)sin(4x)sin(2x)

解答

\int cos (4 x) cos (x) sin (4 x) sin (2 x) d x

\frac{1}{10395} (- 3465 cos^{3} (x) sin (8 x) + 2 (110 cos^{2} (x) (7 sin (9 x) + 9 sin (7 x)) + 5 (11 sin (7 x) - 7 sin (11 x)) + 11 (9 sin (5 x) - 5 sin (9 x)))) + C

求解步骤

\int cos (4 x) cos (x) sin (4 x) sin (2 x) d x

使用三角恒等式改写

= \int \frac{1}{2} sin (8 x) cos (x) sin (2 x) d x

提出常数:

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

= \frac{1}{2} \cdot \int sin (8 x) cos (x) sin (2 x) d x

使用分布积分法

= \frac{1}{2} (- \frac{2}{3} cos^{3} (x) sin (8 x) - \int - \frac{16}{3} cos^{3} (x) cos (8 x) d x)

\int - \frac{16}{3} cos^{3} (x) cos (8 x) d x = - \frac{16}{3} (\frac{1}{2} cos^{2} (x) (\frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + \frac{1}{2} (\frac{1}{18} (\frac{1}{7} sin (7 x) - \frac{1}{11} sin (11 x)) + \frac{1}{14} (\frac{1}{5} sin (5 x) - \frac{1}{9} sin (9 x))))

= \frac{1}{2} (- \frac{2}{3} cos^{3} (x) sin (8 x) - (- \frac{16}{3} (\frac{1}{2} cos^{2} (x) (\frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + \frac{1}{2} (\frac{1}{18} (\frac{1}{7} sin (7 x) - \frac{1}{11} sin (11 x)) + \frac{1}{14} (\frac{1}{5} sin (5 x) - \frac{1}{9} sin (9 x))))))

化简

\frac{1}{2} (- \frac{2}{3} cos^{3} (x) sin (8 x) - (- \frac{16}{3} (\frac{1}{2} cos^{2} (x) (\frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + \frac{1}{2} (\frac{1}{18} (\frac{1}{7} sin (7 x) - \frac{1}{11} sin (11 x)) + \frac{1}{14} (\frac{1}{5} sin (5 x) - \frac{1}{9} sin (9 x)))))) : \frac{1}{10395} (- 3465 cos^{3} (x) sin (8 x) + 2 (110 cos^{2} (x) (7 sin (9 x) + 9 sin (7 x)) + 5 (11 sin (7 x) - 7 sin (11 x)) + 11 (9 sin (5 x) - 5 sin (9 x))))

= \frac{1}{10395} (- 3465 cos^{3} (x) sin (8 x) + 2 (110 cos^{2} (x) (7 sin (9 x) + 9 sin (7 x)) + 5 (11 sin (7 x) - 7 sin (11 x)) + 11 (9 sin (5 x) - 5 sin (9 x))))

解答补常数

= \frac{1}{10395} (- 3465 cos^{3} (x) sin (8 x) + 2 (110 cos^{2} (x) (7 sin (9 x) + 9 sin (7 x)) + 5 (11 sin (7 x) - 7 sin (11 x)) + 11 (9 sin (5 x) - 5 sin (9 x)))) + C

\int sin (2 t) e^{s i n (t)} d t

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\int \frac{2 ln ( x ^{2} )}{x} d x