integral of ((e^{2x}-x^2)(x-1))/(x^2e^x)

解答

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

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

求解步骤

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

乘开

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

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

\int \frac{e ^{x}}{x} d x = Ei (x)

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

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

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

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

化简

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

= \frac{x}{e ^{x}} + \frac{e ^{x}}{x}

解答补常数

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

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