integral of t^2e^{9t}

解答

\int t^{2} e^{9 t} d t

\frac{1}{9} e^{9 t} t^{2} - \frac{2}{729} (9 e^{9 t} t - e^{9 t}) + C

求解步骤

\int t^{2} e^{9 t} d t

使用分布积分法

= \frac{1}{9} e^{9 t} t^{2} - \int \frac{2}{9} e^{9 t} t d t

\int \frac{2}{9} e^{9 t} t d t = \frac{2}{729} (9 e^{9 t} t - e^{9 t})

= \frac{1}{9} e^{9 t} t^{2} - \frac{2}{729} (9 e^{9 t} t - e^{9 t})

解答补常数

= \frac{1}{9} e^{9 t} t^{2} - \frac{2}{729} (9 e^{9 t} t - e^{9 t}) + C

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