1. More U-Substitution: The “Double-U” Substitution with ArcTan(u)
Chapter 5.5
February 13, 2007

2. Techniques of Integration so far…
Use Graph & Area ( )
Use Basic Integral Formulas
Simplify if possible (multiply out, separate fractions…)
Use U-Substitution…..

3. Substitution Rule for Indefinite Integrals
If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then
Substitution Rule for Definite Integrals
If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

4. Evaluate:

5. Compare the two Integrals:
Extra “x”

6. Notice that the extra ‘x’ is the same power as in the substitution:

7. Compare:
Still have an extra “x” that can’t be related to the substitution.
U-substitution cannot be used for this integral

8. Evaluate:
Returning to the original variable “t”:

9. Evaluate:
Returning to the original variable “t”:

10. Evaluate: We have the formula:
Factor out the 9 in the expression 9 + t2:

11. In general: Factor out the a2 in the expression a2 + t2:
We now have the formula:

12. Evaluate:
Returning to the original variable “t”:

13. Use:
It’s necessary to know both forms:
t2 - 2t +26 and 25 + (t-1)2
t2 - 2t +26 = (t2 - 2t + 1) = (t-1)2 + 25

14. Completing the Square:
Comes from

15. How do you know WHEN to complete the square?
Use to solve:
How do you know WHEN to complete the square?
Ans: The equation x2 + x + 3 has NO REAL ROOTS (Check b2 - 4ac)
If the equation has real roots, it can be factored and later we will use Partial Fractions to integrate.

16. Evaluate:

17. Try these:

18. In groups of two/three, use u-substitution to complete: