Evaluate: dx 5-4x - 2x^ 2 . - Vidyadip

Question

Evaluate: $\int\frac{\text{dx}}{\sqrt{5-4\text{x - 2x}^{2}}}.$

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Answer

$\int\frac{\text{dx}}{\sqrt{5-4\text{x}-\text{2x}^{2}}}=\frac{1}{\sqrt{2}}\int\frac{\text{dx}}{\sqrt{\frac{5}{2}-\text{2x - x}^{2}}}=\frac{1}{\sqrt{2}}\int\frac{\text{dx}}{\sqrt{\Bigg(\sqrt\frac{7}{2}\Bigg)^{2}-(\text{x + 1})^{2}}}$
$=\frac{1}{\sqrt{2}}\cdot\sin^{-1}\Bigg(\frac{\text{x + 1}}{\frac{\sqrt{7}}{\sqrt{2}}}\Bigg)+\text{c}$
$\text{Or}\frac{1}{\sqrt{2}}\sin^{-1}\Bigg(\frac{\sqrt{2}}{7}\cdot\text{(x + 1)}\Bigg)+\text{c}.$

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