Question:
Evaluate the following integrals:
$\int e^{\sqrt{x}} d x$
Solution:
Let $I=\int e^{\sqrt{x}} d x$
$\sqrt{\mathrm{X}}=\mathrm{t} ; \mathrm{X}=\mathrm{t}^{2}$
$\mathrm{d} \mathrm{x}=2 \mathrm{tdt}$
$I=2 \int e^{t} t d t$
Using integration by parts,
$I=2\left(t \int e^{t} d t-\int \frac{d}{d t} t \int e^{t} d t\right)$
$=2\left(t e^{t}-\int e^{t} d t\right)$
$=2\left(t e^{t}-e^{t}\right)+c$
$=2 e^{t}(t-1)+c$
Replace the value of $\mathrm{t}$
$=2 e^{\sqrt{x}}(\sqrt{x}-1)+c$
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