Evaluate the following integrals: 1+e^x e^xdx - Vidyadip

Question

Evaluate the following integrals:
$\int\sqrt{1+\text{e}^\text{x}}\text{ e}^\text{x}\text{dx}$

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Answer

$\int\sqrt{1+\text{e}^\text{x}}\text{ e}^\text{x}\text{dx}$
$\text{Let }1+\text{e}^\text{x}=\text{t}$
$\Rightarrow\text{e}^\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}^\text{x}\text{dx}=\text{dt}$
$\text{Now,}\int\sqrt{1+\text{e}^\text{x}}\text{ e}^\text{x}\text{dx}$
$=\int\sqrt{\text{t}}\text{ dt}$
$=\frac{\text{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\text{C}$
$=\frac{2}{3}\text{t}^\frac{3}{2}+\text{C}$
$=\frac{2}{3}(1+\text{e}^\text{x})^\frac{3}{2}+\text{C}$

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