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

解答

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

e^{x} - \frac{x ^{3}}{3} + C

求解步骤

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

乘开

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

= \int e^{x} - x^{2} d x

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 e^{x} d x - \int x^{2} d x

\int e^{x} d x = e^{x}

\int x^{2} d x = \frac{x ^{3}}{3}

= e^{x} - \frac{x ^{3}}{3}

解答补常数

= e^{x} - \frac{x ^{3}}{3} + C

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