Evaluating the indefinite integral $\int\frac{dx}{qx+c}$

Question

Evaluate the indefinite integral (remember to use $\ln |u|$ where appropriate) $$\int\frac{dx}{qx+c}\qquad (q\neq 0) $$

I have no idea how to approach this. But here's what a have so far using the substitution rule:

$$u = qx + C$$

$$dx=\frac{1}cdu $$

Now what?!

Answer 1

By using the substitution:

$$u = qx + c$$

$$du = qdx, dx = \frac{du}{q}$$

You are incorrect in your derivation.

Therefore, we can substitute this:

$$\int\frac{dx}{qx +c} = \int\frac{\frac{du}{q}}{u} = \int\frac{du}{qu} = \frac{1}{q}\int\frac{du}{u} = \frac{1}{q}\ln |u| = \frac{\ln|qx + c|}{q} + C_1$$

Comment if you have questions

Answer 2

$$\int\frac{dx}{qx+c}=\frac{\ln\left| qx+c \right|}{q}+C$$