integral of e^{i7x}cos(5x)

解答

\int e^{i 7 x} cos (5 x) d x

\frac{1}{120} (42 sin (2 x) - 7 sin (12 x) + 24 cos (7 x) sin (5 x)) + i \frac{1}{120} (- 42 cos (2 x) + 7 cos (12 x) + 24 sin (5 x) sin (7 x)) + C

求解步骤

\int e^{i 7 x} cos (5 x) d x

使用分布积分法

= \frac{1}{5} sin (5 x) (cos (7 x) + i sin (7 x)) - \int \frac{1}{5} sin (5 x) (- 7 sin (7 x) + 7 i cos (7 x)) d x

提出常数:

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

= \frac{1}{5} sin (5 x) (cos (7 x) + i sin (7 x)) - \frac{1}{5} \cdot \int sin (5 x) (- 7 sin (7 x) + 7 i cos (7 x)) d x

将

sin (5 x) (- 7 sin (7 x) + 7 i cos (7 x))

改写成标准复数形式:

- 7 sin (5 x) sin (7 x) + 7 sin (5 x) cos (7 x) i

= \frac{1}{5} sin (5 x) (cos (7 x) + i sin (7 x)) - \frac{1}{5} \cdot \int - 7 sin (5 x) sin (7 x) + 7 i sin (5 x) cos (7 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

= \frac{1}{5} sin (5 x) (cos (7 x) + i sin (7 x)) - \frac{1}{5} (- \int 7 sin (5 x) sin (7 x) d x + \int 7 i sin (5 x) cos (7 x) d x)

\int 7 sin (5 x) sin (7 x) d x = \frac{7}{2} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x))

\int 7 i sin (5 x) cos (7 x) d x = i \frac{1}{24} (42 cos (2 x) - 7 cos (12 x))

= \frac{1}{5} sin (5 x) (cos (7 x) + i sin (7 x)) - \frac{1}{5} (- \frac{7}{2} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x)) + i \frac{1}{24} (42 cos (2 x) - 7 cos (12 x)))

化简

\frac{1}{5} sin (5 x) (cos (7 x) + i sin (7 x)) - \frac{1}{5} (- \frac{7}{2} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x)) + i \frac{1}{24} (42 cos (2 x) - 7 cos (12 x))) : \frac{1}{120} (42 sin (2 x) - 7 sin (12 x) + 24 cos (7 x) sin (5 x)) + i \frac{1}{120} (- 42 cos (2 x) + 7 cos (12 x) + 24 sin (5 x) sin (7 x))

= \frac{1}{120} (42 sin (2 x) - 7 sin (12 x) + 24 cos (7 x) sin (5 x)) + i \frac{1}{120} (- 42 cos (2 x) + 7 cos (12 x) + 24 sin (5 x) sin (7 x))

解答补常数

= \frac{1}{120} (42 sin (2 x) - 7 sin (12 x) + 24 cos (7 x) sin (5 x)) + i \frac{1}{120} (- 42 cos (2 x) + 7 cos (12 x) + 24 sin (5 x) sin (7 x)) + C

\frac{\partial}{\partial x} (5 e^{x - y})

\int \frac{3}{( x ^{2} + 1 ) ^{\frac{3}{2}}} d x

\int \frac{2 u}{1 + u ^{2}} d u

y^{^{''}} + y^{^{'}} - 2 y = 2 x

\int 2 e^{u} d u