Results 1 to 2 of 2

Thread: Function Proof

  1. #1
    Apr 2008

    Function Proof

    Prove that if f and g' are continuous real functions on [a,b] then

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Nov 2005
    New York City
    To find $\displaystyle \int_a^b f(g(x))g'(x)dx$ you need to basically compute the primitive and apply the fundamental theorem. Since $\displaystyle f$ is a continous function on $\displaystyle [a,b]$ it has a primitive $\displaystyle F$ so that $\displaystyle F'=f$ on the interval $\displaystyle (a,b)$ and $\displaystyle F$ is continous on $\displaystyle [a,b]$. Say that $\displaystyle g$ is increasing, and consider the function $\displaystyle F(g(x))$ on $\displaystyle [a,b]$. Then, $\displaystyle [F(g(x))]' = f(g(x))g'(x)$ by the Chain rule. Thus, $\displaystyle \int_a^b f(g(x))g'(x) dx = F(g(b)) - F(g(a)) = \int_{g(a)}^{g(b)} f(\xi)d\xi$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof that binomial function is a probability function?
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: Dec 28th 2011, 02:26 PM
  2. Replies: 1
    Last Post: Dec 30th 2010, 04:23 PM
  3. Function proof
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 20th 2009, 10:08 AM
  4. Replies: 0
    Last Post: Sep 14th 2009, 07:13 AM
  5. Another proof using the phi-function
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Apr 29th 2009, 12:28 PM

Search Tags

/mathhelpforum @mathhelpforum