Evaluate: d x 5-8 x-x^2 - Vidyadip

MCQ

Evaluate: $\int \frac{d x}{5-8 x-x^2}$

A
$\frac{1}{\sqrt{21}} \log \left|\frac{\sqrt{21}+x+4}{\sqrt{21}-x-4}\right|+C$
✓
$\frac{1}{2 \sqrt{21}} \log \left|\frac{\sqrt{21}+x+4}{\sqrt{21}-x-4}\right|+C$
C
$\frac{1}{\sqrt{21}} \log \left|\frac{\sqrt{21}-x-4}{\sqrt{21}+x+4}\right|+C$
D
$\frac{1}{2 \sqrt{21}} \log \left|\frac{\sqrt{21}-x-4}{\sqrt{21}+x+4}\right|+C$

✓

Answer

Correct option: B.

$\frac{1}{2 \sqrt{21}} \log \left|\frac{\sqrt{21}+x+4}{\sqrt{21}-x-4}\right|+C$

Let $I=\int \frac{d x}{5-8 x-x^2}=\int \frac{d x}{21-(x+4)^2}$
$=\int \frac{d x}{(\sqrt{21})^2-(x+4)^2}=\frac{1}{2 \sqrt{21}} \log \left|\frac{\sqrt{21}+x+4}{\sqrt{21}-x-4}\right|+C$

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