integral of e^{i7x}cos(4x)

解答

\int e^{i 7 x} cos (4 x) d x

\frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x)) + C

求解步骤

\int e^{i 7 x} cos (4 x) d x

使用分布积分法

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \int \frac{1}{4} sin (4 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}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} \cdot \int sin (4 x) (- 7 sin (7 x) + 7 i cos (7 x)) d x

将

sin (4 x) (- 7 sin (7 x) + 7 i cos (7 x))

改写成标准复数形式:

- 7 sin (4 x) sin (7 x) + 7 sin (4 x) cos (7 x) i

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} \cdot \int - 7 sin (4 x) sin (7 x) + 7 i sin (4 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}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} (- \int 7 sin (4 x) sin (7 x) d x + \int 7 i sin (4 x) cos (7 x) d x)

\int 7 sin (4 x) sin (7 x) d x = \frac{7}{2} (\frac{1}{3} sin (3 x) - \frac{1}{11} sin (11 x))

\int 7 i sin (4 x) cos (7 x) d x = i \frac{1}{66} (- 21 cos (11 x) + 77 cos (3 x))

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} (- \frac{7}{2} (\frac{1}{3} sin (3 x) - \frac{1}{11} sin (11 x)) + i \frac{1}{66} (- 21 cos (11 x) + 77 cos (3 x)))

化简

\frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} (- \frac{7}{2} (\frac{1}{3} sin (3 x) - \frac{1}{11} sin (11 x)) + i \frac{1}{66} (- 21 cos (11 x) + 77 cos (3 x))) : \frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x))

= \frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x))

解答补常数

= \frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x)) + C

\int e^{2 x} x^{3} d x

in v erse l a pl a ce \frac{( s - 1 )}{4 s ^{2} + s + 24}

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

y^{^{'}} = \frac{4 t ^{3} + 1}{2 y - 6}, y (1) = 2

\int \frac{e ^{\frac{6}{t}}}{t ^{2}} d t