A quick question on in logs

Question

I was solving indefinite integrals

$$\int_0^22^x x \,dx$$

I use ILATE as a clue to consider the first function and second function. $2^x$ is a algebraic function or logarithmic function?

Answer 1

For: $$\int\limits_{0}^{2} x\cdot 2^x\:dx$$ Take $2^x =t$, then we get $2^x \cdot \ln(2) \cdot dx=dt$, and, $x=\log_2(t)$. Thus:
\begin{align} \int\limits_{0}^{2} x \cdot 2^x\:dx&=\frac{1}{\ln(2)}\int\limits_{1}^{4}\log_2(t)\: dt \\ &=\frac{1}{(\ln(2))^2}\int\limits_{1}^{4}\ln(t) \: dt \end{align}

Now, I leave it for you. Use by parts (Hint: $\ln(2)= 1\cdot\ln(2)$).