Mathematics

Evaluate $$\displaystyle\int_{0}^{1}xe^{x^{2}}dx$$


SOLUTION
Let $$t={x}^{2}\Rightarrow\,dt=2x\,dx$$

$$\displaystyle\int_{0}^{1}{x{e}^{{x}^{2}}dx}=\dfrac{1}{2}\displaystyle\int_{0}^{1}{2x{e}^{{x}^{2}}dx}$$

When $$x=0\Rightarrow\,t=0$$

When $$x=1\Rightarrow\,t=1$$

$$\dfrac{1}{2}\displaystyle\int_{0}^{1}{{e}^{t}dt}$$

$$=\dfrac{1}{2}\left[{e}^{t}\right]_{0}^{1}$$

$$=\dfrac{1}{2}\left[{e}^{1}-{e}^{0}\right]$$

$$=\dfrac{e-1}{2}$$

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Subjective Medium Published on 17th 09, 2020
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