Evaluate the following integrals: 1 7-6x-x^2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{1}{\sqrt{7-6\text{x}-\text{x}^2}}\text{ dx}$

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Answer

$7 - 6x - x^2$ can be written as $7 - (x^2 + 6x + 9 - 9)$.
Therefore, $7-(\text{x}^2+6\text{x}+9-9)$
$=16-(\text{x}^2+6\text{x}+9)$ $=16-(\text{x}+3)^2$
$=(4)^2-(\text{x}+3)^2$
$\therefore\ \int\frac{1}{\sqrt{7-6\text{x}-\text{x}^2}}\text{ dx}$
$=\int\frac{1}{\sqrt{(4)^2-(\text{x}+3)^2}}\text{ dx}$ Let x + 3 = t $\Rightarrow\text{dx}=\text{dt}$
$\Rightarrow\int\frac{1}{\sqrt{(4)^2-(\text{x}+3)^2}}\text{ dx}=\int\frac{1}{\sqrt{(4)^2-(\text{t})^2}}\text{ dt}$
$=\sin^{-1}\Big(\frac{\text{t}}{4}\Big)+\text{C}$
$=\sin^{-1}\Big(\frac{\text{x}+3}{4}\Big)+\text{C}$

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