Evaluate the following integrals: 1 7-3x-2x^2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{1}{\sqrt{7-3\text{x}-2\text{x}^2}}\text{ dx}$

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Answer

$\int\frac{\text{dx}}{\sqrt{7-3\text{x}-2\text{x}^2}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{7}{2}-\frac{3}{2}\text{x}-\text{x}^2}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{7}{2}-\big(\text{x}^2-\frac{3}{2}\text{x}\big)}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\Big(\frac{\sqrt7}{\sqrt2}\Big)^2-\Big(\text{x}^2+\frac{3}{2}\text{x}+\big(\frac{3}{4}\big)^2-\big(\frac{3}{4}\big)^2\Big)}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\Big(\frac{\sqrt7}{\sqrt2}\Big)^2-\big(\text{x}+\frac{3}{4}\big)^2+\frac{9}{16}}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{7}{2}+\frac{9}{16}-\big(\text{x}+\frac{3}{4}\big)^2}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{56+9}{16}-\big(\text{x}+\frac{3}{4}\big)^2}}$$=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\Big(\frac{\sqrt{65}}{4}\Big)^2-\big(\text{x}+\frac{3}{4}\big)^2}}$
$=\frac{1}{\sqrt2}\sin^{-1}\Bigg[\frac{\text{x}+\frac{3}{4}}{\frac{\sqrt{65}}{4}}{}\Bigg]+\text{C}$ $=\frac{1}{\sqrt2}\sin^{-1}\Big[\frac{4\text{x}+3}{\sqrt{65}}\Big]+\text{C}$

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