Evaluate the following integrals: ^ x dx - Vidyadip

Question

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

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Answer

Let $\text{I}=\int\text{e}^{\sqrt{\text{x}}}\text{dx}$
Let $\sqrt{\text{x}}=\text{t}$
$\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt}$
$\text{I}=2\int\text{e}^{\text{t}}\text{t dt}$
$\text{I}=2[\text{t}\int\text{e}^{\text{t}}\text{dt}-\int(1\int\text{e}^{\text{t}}\text{dt})\text{dt}$
$\text{I}=2[\text{te}^\text{t}-\int\text{e}^{\text{t}}\text{dt}]$
$=2[\text{te}^{\text{t}}-\text{e}^{\text{t}}]+\text{C}$
$=2\text{e}^{\text{t}}(\text{t}-1)+\text{C}$
$\text{I}=2\text{e}^{\sqrt{\text{x}}}(\sqrt{\text{x}}-1)+\text{C}$

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