Indefinite Integration 1 Question 69
70. Evaluate $\int _0^{\pi} \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$.
$(1999,3 M)$
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Solution:
- Let $I=\int _0^{\pi} \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$
$$ \begin{aligned} & =\int _0^{\pi} \frac{e^{\cos (\pi-x)}}{e^{\cos (\pi-x)}+e^{-\cos (\pi-x)}} d x \\ & {\left[\because \int _0^{a} f(x) d x=\int _0^{a} f(a-x) d x\right] } \\ \Rightarrow \quad I & =\int _0^{\pi} \frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}} d x \end{aligned} $$
On adding Eqs. (i) and (ii), we get
$$ \begin{aligned} & =\int _0^{\pi} \frac{e^{\cos x}+e^{-\cos x}}{e^{\cos x}+e^{-\cos x}} d x=\int _0^{\pi} 1 d x=[x] _0^{\pi}=\pi \\ \Rightarrow \quad I & =\pi / 2 \end{aligned} $$
