What is the integral of t^2(e^t)^2 ?

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

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integral of t^2(e^t)^2

integral

t^{2} (e^{t})^{2}

\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^{t})^{2} d t

(e^{t})^{2} = e^{2 t}

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

Apply Integration By Parts

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

= \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

4 x^{10}

\frac{1}{x} + \frac{1}{e ^{x}}

\frac{r ^{2}}{( 1 + r ^{2} )}

\frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )}

\frac{1}{( 3 x ^{2} - 4 x )}

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