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

Question

Evaluate the following integrals:
$\int\sqrt{1+\text{x}-2\text{x}^2}\text{dx}$

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Answer

Let $\text{I}=\int\sqrt{1+\text{x}-2\text{x}^2}\text{dx}$
$=\sqrt{2}\int\sqrt{\frac{1}{2}+\frac{\text{x}}{2}-\text{x}^2}\text{dx}$
$=\sqrt{2}\int\sqrt{\frac{9}{16}-\Big(\frac{1}{16}-\frac{\text{x}}{2}+\text{x}^2\Big)}\text{dx}$
$=\sqrt{2}\int\sqrt{\Big(\frac{3}{4}\Big)^2-\Big(\text{x}-\frac{1}{4}\Big)^2}\text{dx}$
$=\sqrt{2}\begin{Bmatrix}\frac{\Big(\text{x}-\frac{1}{4}\Big)}{2}\sqrt{\frac{1}{2}+\frac{\text{x}}{2}-\text{x}^2}+\frac{9}{32}\sin^{-1}\bigg(\frac{\text{x}-\frac{1}{4}}{\frac{3}{4}}\bigg)\end{Bmatrix}+\text{C}$
$\text{I}=\frac{1}{8}(4\text{x}-1)\sqrt{1+\text{x}-2\text{x}^2}+\frac{9\sqrt{2}}{32}\sin^{-1}\Big(\frac{4\text{x}-1}{3}\Big)+\text{C}$

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