What is the integral of t^2e^{6t} ?

The integral of t^2e^{6t} is 1/6 e^{6t}t^2-1/108 (6e^{6t}t-e^{6t})+C

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integral of t^2e^{6t}

\int t^{2} e^{6 t} d t

\frac{1}{6} e^{6 t} t^{2} - \frac{1}{108} (6 e^{6 t} t - e^{6 t}) + C

Solution steps

\int t^{2} e^{6 t} d t

Apply Integration By Parts

: \frac{1}{6} e^{6 t} t^{2} - \int \frac{1}{3} e^{6 t} t d t

= \frac{1}{6} e^{6 t} t^{2} - \int \frac{1}{3} e^{6 t} t d t

\int \frac{1}{3} e^{6 t} t d t = \frac{1}{108} (6 e^{6 t} t - e^{6 t})

= \frac{1}{6} e^{6 t} t^{2} - \frac{1}{108} (6 e^{6 t} t - e^{6 t})

Add a constant to the solution

= \frac{1}{6} e^{6 t} t^{2} - \frac{1}{108} (6 e^{6 t} t - e^{6 t}) + C

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What is the integral of t^2e^{6t} ?
The integral of t^2e^{6t} is 1/6 e^{6t}t^2-1/108 (6e^{6t}t-e^{6t})+C