Evaluate the following integrals: e^ x^2 =dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x e}^{\text{x}^2}=\text{dx}$

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Answer

Let $\text{I}=\int\text{x e}^{\text{x}^2}=\text{dx}\ ....(1)$

Let $\text{x}^2=\text{t}$ then,

$\text{d}(\text{x}^2)=\text{dt}$

$\Rightarrow2\text{x dx}=\text{dt}$

$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$

Putting $\text{x}^2=\text{t}$ and $\text{x dx}=\frac{\text{dt}}{2}$ in equation (1),

we get,

$\text{I}=\int\text{e}^\text{t}\frac{\text{dt}}{2}$

$=\frac{1}{2}\text{e}^\text{t}+\text{C}$

$=\frac{1}{2}\text{e}^{\text{x}^2}+\text{C}$

$\text{I}=\frac{1}{2}\text{e}^{\text{x}^2}+\text{C}$

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