How to calculate the integral $\int x d(4x) $?

Question

How can I take the following integral?

$$\int x d(4x) $$

It seems very easy. But it is difficult for me.

Is it equal to $4\times \frac12 \times x^2+c$ ?

Answer 1

Yes, of course! It's $$4\int xdx=4\cdot\frac{x^2}{2}+C=2x^2+C$$

Answer 2

$$\int xd(4x)=\frac14\int (4x)d(4x)=\frac14\frac{(4x)^2}{2}+C=2x^2+C$$