∫ X + 1 ( X − 1 ) √ X + 2 D X - Mathematics

Sum
$\int\frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }$

Solution

$\text{ We have,}$
$I = \int \frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}}\text{ dx }$
$\text{ Putting x }+ 2 = t^2$
$\Rightarrow x = t^2 - 2$
$\text{ Diff both sides }$
$dx = 2t \text{ dt }$
$I = \int \frac{\left( t^2 - 2 + 1 \right)2t \text{ dt }}{\left( t^2 - 2 - 1 \right)t}$
$= 2\int \left( \frac{t^2 - 1}{t^2 - 3} \right)dt$
$= 2\int\left( \frac{t^2 - 3 + 2}{t^2 - 3} \right)dt$
$= 2\int \left( \frac{t^2 - 3}{t^2 - 3} \right)dt + 4\int\frac{dt}{t^2 - 3}$
$= 2\int dt + 4\int\frac{dt}{t^2 - \left( \sqrt{3} \right)^2}$
$= 2t + 4 \times \frac{1}{2\sqrt{3}}\text{ log } \left| \frac{t - \sqrt{3}}{t + \sqrt{3}} \right| + C$
$= 2\sqrt{x + 2} + \frac{2}{\sqrt{3}}\text{ log }\left| \frac{\sqrt{x + 2} - \sqrt{3}}{\sqrt{x + 2} + \sqrt{3}} \right| + C$

Concept: Indefinite Integral Problems
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Q 3 | Page 196