What is the integral from 0.1 to 2 of t^2e^{3t} ?

The integral from 0.1 to 2 of t^2e^{3t} is 388.41249…

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integral from 0.1 to 2 of t^2e^{3t}

\int_{0.1}^{2} t^{2} e^{3 t} d t

388.41249 \dots

Solution steps

\int_{0.1}^{2} t^{2} e^{3 t} d t

Apply Integration By Parts

= [0.33333 \dots e^{3 t} t^{2} - \int \frac{2}{3} e^{3 t} t d t]_{0.1}^{2}

\int \frac{2}{3} e^{3 t} t d t = \frac{2}{27} (3 e^{3 t} t - e^{3 t})

= [0.33333 \dots e^{3 t} t^{2} - \frac{2}{27} (3 e^{3 t} t - e^{3 t})]_{0.1}^{2}

Simplify

= [0.33333 \dots e^{3 t} t^{2} - 0.07407 \dots (3 e^{3 t} t - e^{3 t})]_{0.1}^{2}

Compute the boundaries:

388.41249 \dots

= 388.41249 \dots

What is the integral from 0.1 to 2 of t^2e^{3t} ?
The integral from 0.1 to 2 of t^2e^{3t} is 388.41249…