Evaluate the following integrals: 1 2x-x^2 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{1}{\sqrt{2\text{x}-\text{x}^2}}\text{ dx}$

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Answer

$\int\frac{\text{dx}}{\sqrt{2\text{x}-\text{x}^2}}$
$=\int\frac{\text{dx}}{\sqrt{2\text{x}-\text{x}^2-1+1}}$
$=\int\frac{\text{dx}}{\sqrt{1-(\text{x}^2-2\text{x}+1)}}$
$=\int\frac{\text{dx}}{1-(\text{x}-1)^2}$
$=\sin(\text{x}-1)+\text{C}$ $\Big[\because\ \int\frac{\text{dx}}{\sqrt{\text{a}^2-\text{x}^2}}=\sin^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)+\text{C}\Big]$

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