What is the integral of t^2e^{4t} ?

The integral of t^2e^{4t} is 1/4 e^{4t}t^2-1/32 (4e^{4t}t-e^{4t})+C

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integral of t^2e^{4t}

\int t^{2} e^{4 t} d t

\frac{1}{4} e^{4 t} t^{2} - \frac{1}{32} (4 e^{4 t} t - e^{4 t}) + C

Solution steps

\int t^{2} e^{4 t} d t

Apply Integration By Parts

: \frac{1}{4} e^{4 t} t^{2} - \int \frac{1}{2} e^{4 t} t d t

= \frac{1}{4} e^{4 t} t^{2} - \int \frac{1}{2} e^{4 t} t d t

\int \frac{1}{2} e^{4 t} t d t = \frac{1}{32} (4 e^{4 t} t - e^{4 t})

= \frac{1}{4} e^{4 t} t^{2} - \frac{1}{32} (4 e^{4 t} t - e^{4 t})

Add a constant to the solution

= \frac{1}{4} e^{4 t} t^{2} - \frac{1}{32} (4 e^{4 t} t - e^{4 t}) + C

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What is the integral of t^2e^{4t} ?
The integral of t^2e^{4t} is 1/4 e^{4t}t^2-1/32 (4e^{4t}t-e^{4t})+C