Integration by u-Substitution "Millions saw the apple fall, but Newton asked why." -– Bernard Baruch

Objective To integrate by using u-substitution

Recognizing nested derivatives…

What about…

In summary… Pattern recognition: Look for inside and outside functions in integral Determine what u and du would be Take integral Check by taking the derivative!

Change of Variables

Another example

A third example

Guidelines for making a change of variables 1. Choose a u = g(x) 2. Compute du 3. Rewrite the integral in terms of u 4. Evaluate the integral in terms of u 5. Replace u by g(x) 6. Check your answer by differentiating

Try…

Change of variables for definite integrals Thm: If the function u = g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then

First way…

Second way…

Another example (way 1)

Way 2…

Even and Odd functions Let f be integrable on the closed interval [-a,a] If f is an even function, then If f is an odd function then