Evaluate the following definite integral: `int_0^4 1/(sqrt(4x-x^2))dx`

Explanation

Take, ` I=int_{0}^{4} frac{1}{sqrt{4 x-x^{2}}} d x. `
So , `I=int_{0}^{4} frac{1}{sqrt{4 x-x^{2}-4+4}} d x `
`Rightarrow I=int_{0}^{4} frac{1}{sqrt{-(x-2)^{2}+4}} d x Rightarrow I=[sin ^{-1} frac{(x-2)}{2}]_{0}^{4} `
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Rightarrow I=(sin ^{-1} 1-sin ^{-1}(-1)) `
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Rightarrow I=2 sin ^{-1} 1 `
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Rightarrow I=2 frac{pi}{2}=pi `
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