∫ 2 X + 1 ( X − 2 ) ( X − 3 ) D X - Mathematics

Sum
$\int\frac{2x + 1}{\left( x - 2 \right) \left( x - 3 \right)} dx$

Solution

We have,
$I = \int \frac{\left( 2x + 1 \right)dx}{\left( x - 2 \right) \left( x - 3 \right)}$
$\text{Let }\frac{2x + 1}{\left( x - 2 \right) \left( x - 3 \right)} = \frac{A}{x - 2} + \frac{B}{x - 3}$
$\Rightarrow \frac{2x + 1}{\left( x - 2 \right) \left( x - 3 \right)} = \frac{A\left( x - 3 \right) + B\left( x - 2 \right)}{\left( x - 2 \right) \left( x - 3 \right)}$
$\Rightarrow 2x + 1 = A\left( x - 3 \right) + B\left( x - 2 \right)$
$\text{Putting }x - 3 = 0$
$\Rightarrow x = 3$
$\therefore 7 = A \times 0 + B \times \left( 3 - 2 \right)$
$\Rightarrow B = 7$
$\text{Putting }x - 2 = 0$
$\Rightarrow x = 2$
$\therefore 5 = A\left( - 1 \right)$
$\Rightarrow A = - 5$
$\therefore I = - 5\int\frac{dx}{x - 2} + 7\int\frac{dx}{x - 3}$
$= - 5 \log \left| x - 2 \right| + 7 \log \left| x - 3 \right| + C$
$= \log \left| x - 3 \right|^7 - \log \left| x - 2 \right|^5 + C$
$= \log \left| \frac{\left( x - 3 \right)^7}{\left( x - 2 \right)^5} \right| + C$

Concept: Indefinite Integral Problems
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Q 52 | Page 177