How to solve the given integral?

Question

I have the following integral, which is part of a larger function, but this is the only part I'm not sure about how to solve:

$$\int (x\frac{da(x)}{dx})dx$$

The variable a (depending on x) is derived over x, and this derivative is multiplied with x. I want to integrate the entire thing over x. An online integral calculator suggested the result was zero, but I am uncertain as to whether I defined the equation properly in that page.

Could you help me solving this integral?

Answer 1

We have that $\frac {d}{dx}a (x) = a'(x) $. Thus, we get, $$I = \int x a'(x) \mathrm {d}x$$ $$ = x \int a'(x) \mathrm {d}x - \int (\int a'(x) \mathrm {d}x) \frac {d}{dx}(x) \mathrm {d}x$$ $$ = xa (x) - \int a (x) \mathrm {d}x \neq 0$$

We have calculated the integral using integration by parts where $u=x $ and $\mathrm {d}v = a'(x) \mathrm {d}x $. Hope it helps.