# Evaluate the following integrals. intfrac{e^{sin x}}{sec x}dx

Evaluate the following integrals.
$\int \frac{{e}^{\mathrm{sin}x}}{\mathrm{sec}x}dx$
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Alix Ortiz
$\text{The given integral is,}$
$I=\int \frac{{e}^{\mathrm{sin}x}}{\mathrm{sec}x}dx$

$I=\int \mathrm{cos}x\left({e}^{\mathrm{sin}x}\right)dx$
$\text{Using substitution method,}$
${e}^{\mathrm{sin}x}=t$
$\frac{d}{dx}\left({e}^{\mathrm{sin}x}\right)=\frac{dt}{dx}$
Let$\left[\frac{d}{d\left(\mathrm{sin}x\right)}\left({e}^{\mathrm{sin}x}\right)\right]\left[\frac{d}{dx}\left(\mathrm{sin}x\right)\right]=\frac{dt}{dx}$
$\left({e}^{\mathrm{sin}x}\right)\left(\mathrm{cos}x\right)=\frac{dt}{dx}$
$\left({e}^{\mathrm{sin}x}\right)\left(\mathrm{cos}x\right)dx=dt$

$I=\int dt=t+C$
$\text{Putting back the value of t, we get}$
$I={e}^{\mathrm{sin}x}+C$