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受欢迎的微积分 >

integral of e^xcos(ax)

解答

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

解答

\frac{a e ^{x} sin ( a x )}{a ^{2} + 1} + \frac{e ^{x} cos ( a x )}{a ^{2} + 1} + C

求解步骤

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

使用分布积分法

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

使用分布积分法

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

因此

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

整理

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

= \frac{a e ^{x} sin ( a x )}{a ^{2} + 1} + \frac{e ^{x} cos ( a x )}{a ^{2} + 1}

解答补常数

= \frac{a e ^{x} sin ( a x )}{a ^{2} + 1} + \frac{e ^{x} cos ( a x )}{a ^{2} + 1} + C

作图

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