Evaluate: _0^1 e^ x^2 x d x - Vidyadip

Question

Evaluate: $\int_0^1 e^{x^2} x d x$

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Answer

Let, $I=\int_0^1 e^{x^2} x d x$
Put, $x^2=t$
or $x d x=\frac{1}{2} d t$
Also when $x=0$ or $t=0$ and when $x=1 \Rightarrow t=1$
$\begin{aligned} \therefore \quad I & =\frac{1}{2} \int_0^1 e^t d t \\ \text { or } \quad I & =\frac{1}{2}\left[e^t\right]_0^1 \\ & =\frac{1}{2}(e-1)\end{aligned}$

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