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受欢迎的微积分 >

integral of (e^{2x}+e^{5x})/(e^{3x)}

解答

\int \frac{e ^{2 x} + e ^{5 x}}{e ^{3 x}} d x

解答

- \frac{1}{e ^{x}} + \frac{1}{2} e^{2 x} + C

求解步骤

\int \frac{e ^{2 x} + e ^{5 x}}{e ^{3 x}} d x

化简

\frac{e ^{2 x} + e ^{5 x}}{e ^{3 x}} : \frac{1 + e ^{3 x}}{e ^{x}}

= \int \frac{1 + e ^{3 x}}{e ^{x}} d x

使用换元积分法

= \int \frac{1 + u ^{3}}{u ^{2}} d u

乘开

\frac{1 + u ^{3}}{u ^{2}} : \frac{1}{u ^{2}} + u

= \int \frac{1}{u ^{2}} + u d u

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

= \int \frac{1}{u ^{2}} d u + \int u d u

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int u d u = \frac{u ^{2}}{2}

= - \frac{1}{u} + \frac{u ^{2}}{2}

u = e^{x}

代回

= - \frac{1}{e ^{x}} + \frac{( e ^{x} ) ^{2}}{2}

化简

- \frac{1}{e ^{x}} + \frac{( e ^{x} ) ^{2}}{2} : - \frac{1}{e ^{x}} + \frac{1}{2} e^{2 x}

= - \frac{1}{e ^{x}} + \frac{1}{2} e^{2 x}

解答补常数

= - \frac{1}{e ^{x}} + \frac{1}{2} e^{2 x} + C

作图

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