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受欢迎的微积分 >

integral of e^{5x}sin(6x)

解答

\int e^{5 x} sin (6 x) d x

解答

- \frac{6 e ^{5 x} cos ( 6 x )}{61} + \frac{5 e ^{5 x} sin ( 6 x )}{61} + C

求解步骤

\int e^{5 x} sin (6 x) d x

使用分布积分法

= - \frac{1}{6} e^{5 x} cos (6 x) - \int - \frac{5}{6} e^{5 x} cos (6 x) d x

提出常数:

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

= - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} \cdot \int e^{5 x} cos (6 x) d x)

使用分布积分法

= - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} (\frac{1}{6} e^{5 x} sin (6 x) - \int \frac{5}{6} e^{5 x} sin (6 x) d x))

提出常数:

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

= - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} (\frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} \cdot \int e^{5 x} sin (6 x) d x))

因此

\int e^{5 x} sin (6 x) d x = - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} (\frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} \cdot \int e^{5 x} sin (6 x) d x))

整理

\int e^{5 x} sin (6 x) d x

= - \frac{6 e ^{5 x} cos ( 6 x )}{61} + \frac{5 e ^{5 x} sin ( 6 x )}{61}

解答补常数

= - \frac{6 e ^{5 x} cos ( 6 x )}{61} + \frac{5 e ^{5 x} sin ( 6 x )}{61} + C

作图

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