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 $\quad I =\int_0^\pi \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$
Using property $P _5$
$
\begin{array}{l}
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\quad \quad \ldots \ldots(1)
\end{array}
$
Adding eqns (1) and (2),
$
\begin{aligned}
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 \quad I & =\frac{\pi}{2} .
\end{aligned}
$

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