_0^ e^ x e^ x + e^ - x dx - Vidyadip

Question

$\int\limits_0^{\pi} \frac{e^{\cos x}}{e^{\cos{x}}+ e^{-\cos x}} \text{dx}$

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Answer

$\text{I} = \int\limits_0^{\pi} \frac{e^{\cos x}}{e^{\cos{x}}+ e^{-\cos x}} \text{dx} = \int\limits_0^{\pi} \frac{e^{\cos}(\pi - x)}{e^{\cos{(\pi -x)}}+ e^{-\cos (\pi-x})} \text{dx}$
$= \int\limits_0^{\pi} \frac{e^{-\cos x}}{e^{-\cos{x}}+ e^{\cos x}}\text{dx}$
$\text{2I} = \int\limits_0^{\pi} 1. \text{dx} = \bigg[\text{x}\bigg]^{\pi}_{0}$
$= \pi$
$\text{I} = \frac{\pi}{2}$

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