Integration by u-Substitution

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

2. Objective • To integrate by using u-substitution

3. Recognizing nested derivatives…

4. What about…

5. 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!

6. Change of Variables

7. Another example

8. A third example

9. 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

10. Try…

11. 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

12. First way…

13. Second way…

14. Another example (way 1)

15. Way 2…

16. 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