Question

Evaluate the ∫dx/5-3x

Answer 1

Solution.

Make a replacement:

$t=5-3x,$ from here $3x=5-t, x=-\frac{1}{3}t+\frac{5}{3}.$

Then $dx=d(-\frac{1}{3}t+\frac{5}{3})=-\frac{1}{3}dt.$

We will have

$\int{\frac{-\frac{1}{3}dt}{t}}= -\frac{1}{3}\int{\frac{dt}{t}}= -\frac{1}{3}\ln{|t|}+C= -\frac{1}{3}\ln{|{5-3x}|}+C.$

Answer.

$\int{\frac{dx}{5-3x}}= -\frac{1}{3}\ln{|5-3x|}+C$