integral of e^{i8x}cos(5x)

解答

\int e^{i 8 x} cos (5 x) d x

\frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x)) + C

求解步骤

\int e^{i 8 x} cos (5 x) d x

使用分布积分法

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \int \frac{1}{5} sin (5 x) (- 8 sin (8 x) + 8 i cos (8 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 (8 x) + i sin (8 x)) - \frac{1}{5} \cdot \int sin (5 x) (- 8 sin (8 x) + 8 i cos (8 x)) d x

将

sin (5 x) (- 8 sin (8 x) + 8 i cos (8 x))

改写成标准复数形式:

- 8 sin (5 x) sin (8 x) + 8 sin (5 x) cos (8 x) i

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} \cdot \int - 8 sin (5 x) sin (8 x) + 8 i sin (5 x) cos (8 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 (8 x) + i sin (8 x)) - \frac{1}{5} (- \int 8 sin (5 x) sin (8 x) d x + \int 8 i sin (5 x) cos (8 x) d x)

\int 8 sin (5 x) sin (8 x) d x = 4 (\frac{1}{3} sin (3 x) - \frac{1}{13} sin (13 x))

\int 8 i sin (5 x) cos (8 x) d x = i \frac{1}{39} (- 12 cos (13 x) + 52 cos (3 x))

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} (- 4 (\frac{1}{3} sin (3 x) - \frac{1}{13} sin (13 x)) + i \frac{1}{39} (- 12 cos (13 x) + 52 cos (3 x)))

化简

\frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} (- 4 (\frac{1}{3} sin (3 x) - \frac{1}{13} sin (13 x)) + i \frac{1}{39} (- 12 cos (13 x) + 52 cos (3 x))) : \frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x))

= \frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x))

解答补常数

= \frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x)) + C

x \to 0 + lim (x cot (x) csc (x))

t an g e n t x^{4}, at x = 3

\frac{d}{d x} (\frac{16}{( 1 + 2 x ) ^{3}})

d er i v a t i v e arctan (cos (3 x))

\int \frac{x - 1}{x ^{3}} d x