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受欢迎的微积分 >

integral of e^{ax}sin(bx)

解答

\int e^{a x} sin (b x) d x

解答

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

求解步骤

\int e^{a x} sin (b x) d x

使用分布积分法

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

提出常数:

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

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

化简

- a \frac{1}{b} : - \frac{a}{b}

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

使用分布积分法

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

提出常数:

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

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

化简

a \frac{1}{b} : \frac{a}{b}

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

因此

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

整理

\int e^{a x} sin (b x) d x

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

解答补常数

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

作图

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