Evalute : e^ x d x - Vidyadip

Question

Evalute : $\int e^{\sqrt{x}} d x$

✓

Answer

$
\text { Let } I=\int e^{\sqrt{x}} d x
$
Put $\sqrt{x}=t \quad \therefore x=t^2$
$\therefore d x=2 t d t$
$
\begin{aligned}
\therefore I & =\int e^t \cdot 2 t d t=2 \int t e^t d t \\
& =2\left[t \int e^t d t-\int\left\{\frac{d}{d t}(t) \int e^t d t\right\}\right] d t \\
& =2\left[t \cdot e^t-\int 1 \cdot e^t d t\right] \\
& =2\left[t \cdot e^t-e^t\right]+c \\
& =2(t-1) e^t+c \\
& =2(\sqrt{x}-1) e^{\sqrt{x}}+c .
\end{aligned}
$

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