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

Question

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

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Answer

Let $I=\int \frac{d x}{x(5 x-2)}$
Using Partial Fractions: $\frac{1}{x(5 x-2)}$
$=\frac{A}{x}+\frac{B}{5 x-2} 1$
$=A(5 x-$ 2) $+B x$
→ Put $x=0: 1=A(-2) \Longrightarrow A=-1 / 2$
→ Put $x=2 / 5: 1=B(2 / 5) \Longrightarrow B=5 / 2$
Substituting back into the integral:
$I=\int\left(\frac{-1 / 2}{x}+\frac{5 / 2}{5 x-2}\right) d x$
$I=-\frac{1}{2} \ln |x|+\frac{5}{2} \cdot \frac{\ln |5 x-2|}{5}+$ $C$
$I=\frac{1}{2}(\ln |5 x-2|-\ln |x|)+C$
$I=\frac{1}{2} \ln \left|\frac{5 x-2}{x}\right|+C$

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