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受欢迎的微积分 >

integral of e^xcos(9x)

解答

\int e^{x} cos (9 x) d x

解答

\frac{9 e ^{x} sin ( 9 x )}{82} + \frac{e ^{x} cos ( 9 x )}{82} + C

求解步骤

\int e^{x} cos (9 x) d x

使用分布积分法

= e^{x} \frac{1}{9} sin (9 x) - \int e^{x} \frac{1}{9} sin (9 x) d x

提出常数:

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

= e^{x} \frac{1}{9} sin (9 x) - \frac{1}{9} \cdot \int e^{x} sin (9 x) d x

使用分布积分法

= e^{x} \frac{1}{9} sin (9 x) - \frac{1}{9} (- \frac{1}{9} e^{x} cos (9 x) - \int - \frac{1}{9} e^{x} cos (9 x) d x)

提出常数:

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

= e^{x} \frac{1}{9} sin (9 x) - \frac{1}{9} (- \frac{1}{9} e^{x} cos (9 x) - (- \frac{1}{9} \cdot \int e^{x} cos (9 x) d x))

因此

\int e^{x} cos (9 x) d x = e^{x} \frac{1}{9} sin (9 x) - \frac{1}{9} (- \frac{1}{9} e^{x} cos (9 x) - (- \frac{1}{9} \cdot \int e^{x} cos (9 x) d x))

整理

\int e^{x} cos (9 x) d x

= \frac{9 e ^{x} sin ( 9 x )}{82} + \frac{e ^{x} cos ( 9 x )}{82}

解答补常数

= \frac{9 e ^{x} sin ( 9 x )}{82} + \frac{e ^{x} cos ( 9 x )}{82} + C

作图

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