Do you never need the chain rule when integrating?

[lLovePhysics]

Here are some questions I have concerning AP Calculus that I have compiled while doing my homework assignments. Please help me answer them. Thank you very much!

Questions:

-Do you never need to worry about the chain rule when integrating?

I'll add on to these questions when I have more.

 

[HallsofIvy]

Depends on what you mean. Strictly speaking the "chain rule" is a rule for differentiation so, no you don't have to worry about it when integrating.

However, since integration is the opposite of differentiating, each rule for differentiating has an "opposite" that is used for integrating. The opposite of the "chain rule" is "substitution".

For example, the derivative of sin(x²) is, using the chain rule, cos(x²)(2x). In order to integrate $\int cos(x^2)(2x dx)$, you would let u= x^2 so du= 2xdx and the integral becomes $\int cos(u) du= sin(u)+ C= sin(x^2)+ C$. An important difference is that, since you cannot move variables in or out of the integral that "2x" or at least the "x" has to already be in the integral.

If I had $\int xcos(x^2)dx$ I could write it as $(1/2)(2)\int cos(x^2)(xdx)= (1/2)\int cos(x^2)(2xdx)$ and use the substitution $u= x^2$. If I had $\int cos(x^2)dx$, I'm stuck- I can't move the missing "x" into the integration.

 

[Gib Z]

We actually do use the chain rule for differentiation when doing integrals with substitutions, exactly the same way we use it when differentiating =] $\int \cos (x^2) 2x dx \to \int \cos u du/dx \cdot dx$.