MAT237Y1 Lecture Notes - Quotient Rule, Product Rule
Document Summary
As i mentioned before, there is no such thing as a product rule or a quotient rule for integrals. This means we need another way to solve that stu , which in short, means extra work. Think back to derivatives and the chain rule. Well, remember, the integral is just everything backwards, so. F (cid:48)(g(x))g(cid:48)(x) dx = f (g(x)) + c (cid:90) (cid:90) But recall from before that f (cid:48)(x) = f(x), so this all comes down to f(g(x))g(cid:48)(x) dx = f (g(x)) + c. Then du dx du = g(cid:48)(x)dx (think back to di erentials and how we treated dx like a variable there!). Example: evaluate (cid:90) f(u)du = f (u) + c (cid:90) (cid:112)1 + x2 dx. The idea is that we need to let u be something. You can certainly do trial and error, but the catch is you have to cancel out all the x"s.
