 # Evaluate the following definite integrals: int_0^1(x e^(-x^2+2))dx iohanetc 2021-02-19 Answered
Evaluate the following definite integrals:
${\int }_{0}^{1}\left(x{e}^{-{x}^{2}+2}\right)dx$
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Step 1
To Evaluate the following definite integrals:
Step 2
Given That
${\int }_{0}^{1}x{e}^{-{x}^{2}+2}dx$
Let ${e}^{-{x}^{2}+2}=u$
$d\frac{u}{dx}=-2x\left({e}^{-{x}^{2}+2}\right)$
$d\frac{u}{-2}=x{e}^{-{x}^{2}+2}dx$
Hence ${\int }_{0}^{1}x{e}^{-{x}^{2}+2}dx={\int }_{0}^{1}d\frac{u}{-2}$
$=-\frac{1}{2}{\left[4\right]}_{0}^{1}$
Since we have $u={e}^{-{x}^{2}+2}$ Then we have
$=-\frac{1}{2}{\left[{e}^{-{x}^{2}+2}\right]}_{0}^{1}$
$=-\frac{1}{2}\left[{e}^{1}-{e}^{2}\right]$
$=-\frac{1}{2}\left[e-{e}^{2}\right]=\frac{1}{2}\left[{e}^{2}-e\right]$
$\because {\int }_{0}^{1}x{e}^{-{x}^{2}+2}dx=\frac{1}{2}\left({e}^{2}-e\right)$