Evaluate: 5 x+2 3 x-4 d x - Vidyadip

Question

Evaluate:
$\int \frac{5 x+2}{3 x-4} \cdot d x$

✓

Answer

$\int \frac{5 x+2}{3 x-4} d x$
$=\int \frac{\frac{5}{3}(3 x-4)+\frac{20}{3}+2}{3 x-4} d x$
$=\int \frac{\frac{5}{3}(3 x-4)+\frac{26}{3}}{3 x-4} d x$
$=\int\left[\frac{5}{3}+\frac{\left(\frac{26}{3}\right)}{3 x-4}\right] d x$
$=\frac{5}{3} \int 1 d x+\frac{26}{3} \int \frac{1}{3 x-4} d x$
$=\frac{5 x}{3}+\frac{26}{3} \cdot \frac{1}{3} \log |3 x-4|+c$
$=\frac{5 x}{3}+\frac{26}{9} \log |3 x-4|+c .$

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