Question

√1+sin2x/1+cos2x eˣ ,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

HELLO DEAR,

GIVEN:-
$\bold{\int{{\sqrt{1 + sin2x}\over {1 + cos2x}e^x}\,dx}}$

$\bold{=\int{{\sqrt{(sinx + cosx)^2} \over {2cos^2x}}e^x}\,dx}$

$\bold{= \int{{(sinx + cosx) \over 2cos^2x}e^x}\,dx}$

$\bold{= {1 \over 2}\int{e^x[tanxsecx + secx]}\,dx}$

we know if integral is in the form of e^x[f(x) + f'(x)].dx so, your integral is e^x.f(x) + C.

therefore, $\bold{I = (1/2)e^x(secx) + C}$

I HOPE ITS HELP YOU DEAR,
THANKS