integral of (5x-7)e^{5x-7}

解答

\int (5 x - 7) e^{5 x - 7} d x

e^{5 x - 7} x - \frac{8}{5} e^{5 x - 7} + C

求解步骤

\int (5 x - 7) e^{5 x - 7} d x

乘开

(5 x - 7) e^{5 x - 7} : 5 e^{5 x - 7} x - 7 e^{5 x - 7}

= \int 5 e^{5 x - 7} x - 7 e^{5 x - 7} 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 5 e^{5 x - 7} x d x - \int 7 e^{5 x - 7} d x

\int 5 e^{5 x - 7} x d x = e^{5 x - 7} x - \frac{1}{5} e^{5 x - 7}

\int 7 e^{5 x - 7} d x = \frac{7}{5} e^{5 x - 7}

= e^{5 x - 7} x - \frac{1}{5} e^{5 x - 7} - \frac{7}{5} e^{5 x - 7}

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