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

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

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integral of t^2e^{2t}

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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