Question

hey guys,
I am studying integration by substitution, and I get the basic method, and understand why the integral of f(g(x))*g'(x) dx is the same as f(u) du. (Given that u = g(x), meaning that dx = du/g'(x)).
However, how would you integrate f(g(x) dx?
When I try, I go through the basic method of letting u = g(x), then finding dx in terms of du, which is always du/g'(x). However, since we have f(g(x)), not f(g(x))*g'(x), the g'(x) in the denominator has nothing to cancel with, leaving you with the integral of f(u)/g'(x) du.
This is just as complex, so can someone show me where Ihave gone wrong?

Answer 1

In general, I see no one way to integrate things of the form:
\[\int\limits_{}^{}f(g(x)) dx\]

Answer 2

generally you should find x=h(u) then it is going to be dx=h'(u)du and all integran are related to u.

Answer 3

It will vary depending on what f composed with g equals

Answer 4

I like what @mahmit2012 said.

Answer 5

suppose g(x)=u ->x=h(u) -> g(h(u))=u.

Answer 6

so you have int(f(u)h'(u)du

Answer 7

h=g^-1

Answer 8

so would you then integrate by parts for u?

Answer 9

take a look to:
http://openstudy.com/users/mahmit2012#/updates/4ffe3c7be4b09082c06f8009

Answer 10

at which question?

Answer 11

whatever you need.
Go to method 1,2 and 3