integral of ((e^{x/2}-e^{-x/2})^2)/4

解答

\int \frac{( e ^{\frac{x}{2}} - e ^{- \frac{x}{2}} ) ^{2}}{4} d x

\frac{1}{4} (e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)) + C

求解步骤

\int \frac{( e ^{\frac{x}{2}} - e ^{- \frac{x}{2}} ) ^{2}}{4} d x

提出常数:

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

= \frac{1}{4} \cdot \int (e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} d x

使用分布积分法

= \frac{1}{4} ((e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - \int (e^{x} - e^{- x}) x d x)

\int (e^{x} - e^{- x}) x d x = 2 x cosh (x) - 2 sinh (x)

= \frac{1}{4} ((e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - (2 x cosh (x) - 2 sinh (x)))

乘开

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

= \frac{1}{4} (e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x))

解答补常数

= \frac{1}{4} (e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)) + C

\int \frac{x ^{2}}{x ^{3} - 1} d x

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