Mathematics - Indefinite Integration Question with Solution | TestHub

Given$f\left(\frac{3x-4}{3x+4}\right)=x+2, x\ne -\frac{4}{3}$

Let,$\frac{3x-4}{3x+4}=t$

$3x-4=3tx+4t$

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

$\therefore f\left(t\right)=\frac{10-2t}{3-3t}$

$\Rightarrow f\left(x\right)=\frac{2x-10}{3x-3}$

$\int f\left(x\right)dx=\int \frac{2x-10}{3x-3}dx$

$=\int \frac{2x}{3x-3}dx-10\int \frac{dx}{3x-3}$

$=\frac{2x}{3}-\frac{8}{3}\ln \left|x-1\right|+C$, where$C$is the constant of integration.