Evaluate the indefinite integral $\int\frac{dx}{x\ln\left(7x\right)}$

Question

Evaluate the indefinite integral. I am having trouble.

$$\int\frac{dx}{x\ln\left(7x\right)}$$

Help me. Please!

Answer 1

Hint:

Let $u=\ln 7x$, then $du=\frac 1xdx$.

Answer 2

$$\int\frac{dx}{x\ln7x}=\int\frac{dx}{x}\cdot\frac{1}{\ln7x}=|\ln7x=t\Rightarrow \frac{t}{7}dt=\frac{dx}{x}|$$ $$=\frac{1}{7}\int\frac{dt}{t}=\frac{1}{7}\ln |t|=\frac{1}{7}\ln |\ln7x|+C$$

Answer 3

Directly: Using that

$$\int \frac{f'(x)}{f(x)}dx=\log f(x)+C$$

You have

$$(\log 7x)'=\frac1x\implies\int\frac{dx}{x\log 7x}=\int\frac{(\log 7x)'}{\log 7x}dx=\log\log 7x+C$$