Mathematics

$$\displaystyle \int_{0}^{1}xe^{x}\ dx=$$


ANSWER

$$1$$


SOLUTION
$$\displaystyle\int_{0}^{1}{x{e}^{x}dx}$$

Integrating by parts,

Let $$u=x\Rightarrow\,du=dx$$

$$dv={e}^{x}dx\Rightarrow\,v={e}^{x}$$

$$\displaystyle\int_{0}^{1}{x{e}^{x}dx}=\left[x{e}^{x}\right]_{0}^{1}-\displaystyle\int_{0}^{1}{{e}^{x}dx}$$

$$=\left[{e}^{1}-0\right]_{0}^{1}-\left[{e}^{x}\right]_{0}^{1}$$

$$=e-\left[e-{e}^{0}\right]=e-e+1=1$$
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Single Correct Medium Published on 17th 09, 2020
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