∫ 1 4 X 2 + 12 X + 5 D X - Mathematics

Sum
$\int\frac{1}{4 x^2 + 12x + 5} dx$

Solution

$\int\frac{dx}{4 x^2 + 12x + 5}$
$= \frac{1}{4}\int\frac{dx}{x^2 + 3x + \frac{5}{4}}$
$= \frac{1}{4}\int\frac{dx}{x^2 + 3x + \left( \frac{3}{2} \right)^2 - \left( \frac{3}{2} \right)^2 + \frac{5}{4}}$
$= \frac{1}{4}\int\frac{dx}{\left( x + \frac{3}{2} \right)^2 - \frac{9}{4} + \frac{5}{4}}$
$= \frac{1}{4}\int\frac{dx}{\left( x + \frac{3}{2} \right)^2 - 1^2}$
$\text{ let x} + \frac{3}{2} = t$
$\Rightarrow dx = dt$
$Now, \frac{1}{4}\int\frac{dx}{\left( x + \frac{3}{2} \right)^2 - 1^2}$
$= \frac{1}{4}\int\frac{dx}{t^2 - 1^2}$
$= \frac{1}{4} \times \frac{1}{2 \times 1} \text{ log }\left| \frac{t - 1}{t + 1} \right| + C$
$= \frac{1}{8} \text{ log }\left| \frac{x + \frac{3}{2} - 1}{x + \frac{3}{2} + 1} \right| + C$
$= \frac{1}{8} \text{ log }\left| \frac{x + \frac{1}{2}}{x + \frac{5}{2}} \right| + C$
$= \frac{1}{8} \text{ log }\left| \frac{2x + 1}{2x + 5} \right| + C$

Concept: Indefinite Integral Problems
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.15 | Q 1 | Page 86