Evaluate the following definite integrals: _ 0 ^21 3+2x-x^2 dx - Vidyadip

Question

Evaluate the following definite integrals:$\int_{0}^\limits{2}\frac{1}{\sqrt{3+2\text{x}-\text{x}^2}}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{2}\frac{1}{\sqrt{3+2\text{x}-\text{x}^2}}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{2}\frac{1}{\sqrt{-\text{x}^2+2\text{x}-1+1+3}}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{2}\frac{1}{\sqrt{-(\text{x}-1)^2+4}}\text{ dx}$
$\Rightarrow\text{I}=\Big[\sin^{-1}\frac{(\text{x}-1)}{2}\Big]^{2}_0$
$\Rightarrow\text{I}=\sin^{-1}\frac{1}{2}-\sin^{-1}\Big(-\frac{1}{2}\Big)$
$\Rightarrow\text{I}=2\sin^{-1}\frac{1}{2}$
$\Rightarrow\text{I}=2\times\frac{\pi}{6}$
$\Rightarrow\text{I}=\frac{\pi}{3}$

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