-4-4dx-1-cos 2x is equal to

2∫_0^π/41/2cos^2xdx=∫_0^π/4sec^2x=tanx=1 ...

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Inequalities of Integrals

∫-π/4π/4dx/1+...

Question

$\int_{\frac{- π}{4}}^{\frac{π}{4}} \frac{d x}{1 + c o s 2 x}$ is equal to

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\int_{- π / 4}^{π / 4} \frac{d x}{1 + cos 2 x}

I = \frac{2}{π} \frac{π}{4} \int - \frac{π}{4} \frac{d x}{(1 + e^{sin x}) (2 - cos 2 x)},

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π / 4 \int - π / 4 \frac{d x}{1 + sin x}

I = \frac{2}{π} \frac{π}{4} \int - \frac{π}{4} \frac{d x}{(1 + e^{sin x}) (2 - cos 2 x)}

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I = \frac{2}{π} \frac{π}{4} \int - \frac{π}{4} \frac{d x}{(1 + e^{sin x}) (2 - cos 2 x)}

27 I^{2}

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