What is the integral of t^2e^{7t} ?

The integral of t^2e^{7t} is 1/7 e^{7t}t^2-2/343 (7e^{7t}t-e^{7t})+C

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integral of t^2e^{7t}

\int t^{2} e^{7 t} d t

\frac{1}{7} e^{7 t} t^{2} - \frac{2}{343} (7 e^{7 t} t - e^{7 t}) + C

Solution steps

\int t^{2} e^{7 t} d t

Apply Integration By Parts

: \frac{1}{7} e^{7 t} t^{2} - \int \frac{2}{7} e^{7 t} t d t

= \frac{1}{7} e^{7 t} t^{2} - \int \frac{2}{7} e^{7 t} t d t

\int \frac{2}{7} e^{7 t} t d t = \frac{2}{343} (7 e^{7 t} t - e^{7 t})

= \frac{1}{7} e^{7 t} t^{2} - \frac{2}{343} (7 e^{7 t} t - e^{7 t})

Add a constant to the solution

= \frac{1}{7} e^{7 t} t^{2} - \frac{2}{343} (7 e^{7 t} t - e^{7 t}) + C

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What is the integral of t^2e^{7t} ?
The integral of t^2e^{7t} is 1/7 e^{7t}t^2-2/343 (7e^{7t}t-e^{7t})+C