Question

Is possible to solve the integral of 1/(a/x -b) . If yes, which is the solution?

Answer 1

\[\int \frac 1{\frac ax - b}\]
right?

Answer 2

Yes

Answer 3

you can simplify it to \[\int \frac 1{\frac{a-bx}x}\]
and then \[\int \frac x{a-bx}\]
does that help you see the integration now?

Answer 4

ok, do you have the solution for this?

Answer 5

yes

Answer 6

Would you mind posting it?

Answer 7

Substitution: \(u=a-bx~\Rightarrow~du=-dx\). Also, \(u=a-bx~\iff~x=\dfrac{a-u}{b}\).
So you have
\[\begin{align*}\int\frac{x}{a-bx}~dx&=-\int\frac{\frac{a-u}{b}}{u}~du\\
&=\int\frac{u-a}{bu}~du\\
&=\int\left(\frac{1}{b}-\frac{a}{bu}\right)~du
\end{align*}\]