Evaluate the following definite integrals: _ 0 ^ 1 1+ dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\frac{1}{1+\sin\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{\pi}\frac{1}{1+\sin\text{x}}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{\pi}\frac{1-\sin\text{x}}{(1+\sin\text{x})(1-\sin\text{x})}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{\pi}\frac{1-\sin\text{x}}{1+\sin^2\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{\pi}\frac{1-\sin\text{x}}{\cos^2\text{x}}\text{ dx}$ $\big[\because\sin^2\text{x}+\cos^2\text{x}=1\big]$
$\Rightarrow\text{I}=\int_{0}^\limits{\pi}\sec^2\text{x}-\sec\text{x}\tan\text{x}\text{ dx}$
$\Rightarrow\text{I}=\big[\tan\text{x}-\sec\text{x}\big]^{\pi}_0$
$\Rightarrow\text{I}=(\tan\pi-\sec\pi)-(\tan0-\sec0)$
$\Rightarrow\text{I}=0+1-(0-1)$
$\Rightarrow\text{I}=1+1$
$\Rightarrow\text{I}=2$

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