Evaluate _0^ e^ x e^ x +e^ - x d x. - Vidyadip

Question

Evaluate $\int_0^\pi \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$.

✓

Answer

Let $ I =\int_0^\pi \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$
Using property $P _5$
$I=\int_0^\pi \frac{e^{\cos (\pi-x)}}{e^{\cos (\pi-x)}+e^{-\cos (\pi-x)}} d x$
$I=\int_0^\pi \frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}} d x \ldots \ldots(1)$
Adding eqns $(1)$ and $(2),$
$2 I =\int_0^\pi \frac{\left(e^{\cos x}+e^{-\cos x}\right)}{\left(e^{\cos x}+e^{-\cos x}\right)} d x$
$ =\int_0^\pi d x=(x)_0^\pi=\pi-0$
$2 I =\pi$
$\therefore I =\frac{\pi}{2} .$

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