Question

Solve:

$\int \frac{2tdt}{3t+4}$

Answer 1

$\int \frac{2tdt}{3t+4}=\frac{2}{3}\int \frac{3t}{3t+4}dt$

$=\frac{2}{3}\int \frac{3t+4-4}{3t+4}dt$

$=\frac{2}{3}\int \frac{3t+4}{3t+4}dt-\frac{2}{3}\int \frac{4}{3t+4}dt$

$=\frac{2}{3}\int dt-\frac{8}{3}\int \frac{dt}{3t+4}$

$=\frac{2}{3}t-\frac{8}{3}log(3t+4).\frac{1}{3}+c$

$=\frac{2t}{3}-\frac{8}{9}log(3t+4)+c$