Integration involving $\log_2(x)$

Question

Having a hard time going about this problem:

$$\int{\frac{\ln(2)\log_2(x)}{x}}$$

I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to:

$$\ln(2)\int{\dfrac{\log_2(x)}{x}}$$

That's about all I've got.

Answer 1

Since $\log_2(x)=\frac{\ln x}{\ln 2}$ then your anti-derivative becomes

$$\int\frac{\ln x}{x}dx=\int\frac1x\times\ln xdx=\frac12(\ln x)^2+C $$