Question

I need help understanding a basic theorem of calculus - why does the g'(x) disappear when taking the integral f(g(x)) g'(x) dx = F(g(x)) + C

Answer 1

For example, when taking the integral of (2x+1)(x^2+x) dx, u = (x^2+x) and du = (2x+1) dx, which gives us integral u du = [u^2 / 2] + C= [(x^2+x)^2 / 2] + C

Answer 2

the derivative of f(x) = f'(x) dx

oftentimes, the dx is forgotten or omitted on purpose.

the same way that the integral of f'(x) dx = f(x), not simply integral of f'(x) = fx. dx is very important.

not sure if this helps

Answer 3

the examples of integrals you gave are very convenient and simple cases where what you're integrating is already in its convenient u' du form and ready to integrate