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受欢迎的微积分 >

integral of (e^{x/a}+e^{-x/a})^2

解答

\int (e^{\frac{x}{a}} + e^{- \frac{x}{a}})^{2} d x

解答

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

求解步骤

\int (e^{\frac{x}{a}} + e^{- \frac{x}{a}})^{2} d x

使用换元积分法

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

提出常数:

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

= a \cdot \int \frac{( u ^{2} + 1 ) ^{2}}{u ^{3}} d u

乘开

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

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

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

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

= a (\frac{u ^{2}}{2} + 2 ln ∣ u ∣ - \frac{1}{2 u ^{2}})

u = e^{\frac{x}{a}}

代回

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

化简

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

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

解答补常数

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

作图

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