integral of e^{6x}(x^2-11x+1)

解答

\int e^{6 x} (x^{2} - 11 x + 1) d x

\frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x} + C

求解步骤

\int e^{6 x} (x^{2} - 11 x + 1) d x

乘开

e^{6 x} (x^{2} - 11 x + 1) : e^{6 x} x^{2} - 11 e^{6 x} x + e^{6 x}

= \int e^{6 x} x^{2} - 11 e^{6 x} x + e^{6 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 e^{6 x} x^{2} d x - \int 11 e^{6 x} x d x + \int e^{6 x} d x

\int e^{6 x} x^{2} d x = \frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x})

\int 11 e^{6 x} x d x = \frac{11}{36} (6 e^{6 x} x - e^{6 x})

\int e^{6 x} d x = \frac{1}{6} e^{6 x}

= \frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x}) - \frac{11}{36} (6 e^{6 x} x - e^{6 x}) + \frac{1}{6} e^{6 x}

化简

\frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x}) - \frac{11}{36} (6 e^{6 x} x - e^{6 x}) + \frac{1}{6} e^{6 x} : \frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x}

= \frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x}

解答补常数

= \frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x} + C

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