Question

Can anyone explain this substitution rule to me?

http://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables_files/eq0001MP.gif

Answer 1

Think back to the chain rule.
\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\)

Answer 2

Sure,
here u=g(x) is simply substituted in the given integral,
u=g(x), take derivative w.r.t x on both sides, you will get
du/dx = dg(x)/dx i.e. du/dx = g'(x), so this will be simply equal to du=g'(x)dx
now substitute u and du in the given equation and u will understand.

Answer 3

Note that \(c=g(a)\), \(d=g(b)\).

Answer 4

further more as u can notice, the limits a,b are changed to d,c!
This is because now the variable has been changed from x->u
So obviously the values of the limits will also be changed.
@oldrin.bataku i was about to write that :)

Answer 5

cheers guys! much simpler than i thought!