Example one... http://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithRoots.aspx.... They use U sub with out canceling out the whole term... please explain

Question

phi · Answer

are you asking about
$$\int \frac{x+2}{\sqrt[3]{x-3}} dx $$

anonymous · Answer

$$\int\limitsegrate{\frac{x+2}{cuberoot({x-3)}}}$$

phi · Answer

which part is confusing?

you agree  we can say  $$ u^3 = x-3 $$
and 
$$ x= u^3+3 \text{  and } dx = 3u^2 du $$

anonymous · Answer

lol your latex is better... but that is the correct integrand

anonymous · Answer

the dx=3u^2du doesnt cancel which in my past has shown that it cannot be used

phi · Answer

It does not have to cancel.  We are making a substitution, and the new integral is the old integral with different variables.

the top   x+2  becomes u^3 +5
the bottom becomes u
the dx becomes  3 u^2 du
$$ \int \frac{u^3+5}{u} 3 u^2 du = \int (u^3+5) 3u du=\int 3u^4+15u \ du $$

phi · Answer

to be complete we should change the limits of the integration.

phi · Answer

The question is "why do you think it has to cancel?"  That is a "fact" that is not true.

anonymous · Answer

Becasue i relate u sub to the chain rule, so inorder to use it i imagine that in order to use it you have to have the derivative (du) in the origional function

anonymous · Answer

Is this a regular use of u sub

anonymous · Answer

Specificially  my problem is that the u^2 seems to be created out of thin air, when the derivative of x is taken