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Evaluation of definite integrals

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IF $I = \frac{2}{π} \int_{- π / 4}^{π / 4} \frac{d x}{(1 + e^{\sin x}) (2 - \cos 2 x)}$ then the value of $27 I^{2}$ is

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Solution

$\begin{array}{l} I = \frac{2}{π} \int_{- π / 4}^{π / 4} \frac{d x}{(1 + e^{\sin x}) (2 - \cos 2 x)} \\ I = \frac{2}{π} \int_{- π / 4}^{π / 4} \frac{e^{\sin x}}{(1 + e^{\sin x}) (2 - \cos 2 x)} d x (\sin c e \int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x) \\ Adding the above both integrals \\ 2I= \frac{2}{π} \int_{- π / 4}^{π / 4} \frac{d x}{(2 - \cos 2 x)} \\ I = \frac{1}{π} \int_{- π / 4}^{π / 4} \frac{\sec^{2} x d x}{1 + 3 \tan^{2} x} \\ = \frac{2}{3 \sqrt{3}} \\ 27 I^{2} = 4 \end{array}$

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