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:
∫(xda(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
We have that ddxa(x)=a′(x). Thus, we get, I=∫xa′(x)dx =x∫a′(x)dx−∫(∫a′(x)dx)ddx(x)dx =xa(x)−∫a(x)dx≠0
We have calculated the integral using integration by parts where u=x and dv=a′(x)dx. Hope it helps.
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