integral of x^3ye^{xy}+3x^2e^{xy}

解答

\int x^{3} y e^{x y} + 3 x^{2} e^{x y} d x

3 (e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})) + e^{y x} x^{3} - \frac{3 ( y ^{2} e ^{y x} x ^{2} - 2 ( y e ^{y x} x - e ^{y x} ) )}{y ^{3}} + C

求解步骤

\int x^{3} y e^{x y} + 3 x^{2} e^{x y} 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 3 e^{y x} x^{2} d x + \int y e^{y x} x^{3} d x

\int 3 e^{y x} x^{2} d x = 3 (e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}}))

\int y e^{y x} x^{3} d x = e^{y x} x^{3} - \frac{3 ( y ^{2} e ^{y x} x ^{2} - 2 ( y e ^{y x} x - e ^{y x} ) )}{y ^{3}}

= 3 (e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})) + e^{y x} x^{3} - \frac{3 ( y ^{2} e ^{y x} x ^{2} - 2 ( y e ^{y x} x - e ^{y x} ) )}{y ^{3}}

解答补常数

= 3 (e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})) + e^{y x} x^{3} - \frac{3 ( y ^{2} e ^{y x} x ^{2} - 2 ( y e ^{y x} x - e ^{y x} ) )}{y ^{3}} + C

\int \frac{2 x + 5}{x ^{2} + 1} d x

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