What is the integral of te^{t+2} ?

The integral of te^{t+2} is e^2(e^tt-e^t)+C

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integral of te^{t+2}

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

e^{2} (e^{t} t - e^{t}) + C

Solution steps

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

t e^{t + 2} = e^{2} t e^{t}

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

: e^{t} t - \int e^{t} d t

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

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

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

Add a constant to the solution

= e^{2} (e^{t} t - e^{t}) + C

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