Evaluate the following integrals: ^1_0xe^ x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^1_0\text{xe}^{\text{x}^2}\text{dx}$

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Answer

We have,
$\text{I}=\int\limits^1_0\text{xe}^{\text{x}^2}\text{dx}=\frac{1}{2}\int\limits^1_0\text{e}^{\text{x}^2}2\text{dx}$
Putting $\text{x}^2=\text{z}$
$2\text{x dx}=\text{dz}$
When $\text{x}\rightarrow0;\text{ z}\rightarrow0$
And $\text{x}\rightarrow1;\text{ z}\rightarrow1$
$\therefore\ \text{I}=\frac{1}{2}\int\limits^1_0\text{e}^2\text{ dz}$
$=\frac{1}{2}\times\big[\text{e}^{\text{z}}\big]^1_0$
$=\frac{1}{2}(\text{e}-\text{e}^0)$
$=\frac{1}{2}(\text{e}-1)$

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