Question

Evaluate:
${\int }_0^{\frac{\pi }{4}}{e}^{\sin x}(1+\tan x\sec x)dx$

Answer 1

given, ${\int }_0^{\frac{\pi }{4}}{e}^{\sin x}(1+\tan x\sec x)dx$

Let $e^{\sin x}\sec x=z$

or, $dz=(e^{\sin x}.\cos x.\sec x+e^{\sin x}\sec x.\tan x)dx$

or, $dz=e^{\sin x}(1+\sec x.\tan x)dx$

So the given integration will take the form

$\Rightarrow$ ${\int }_0^{\frac{\pi }{4}}d\left(e^{\sin x}\sec x\right)$

$={\left[e^{\sin x}\sec x\right]}_0^{\frac{\pi }{4}}$

$=(\sqrt{2}{e}^{\frac{1}{\sqrt{2}}}-1)$