More U-Substitution: The “Double-U” Substitution with ArcTan(u)

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More U-Substitution: The “Double-U” Substitution with ArcTan(u)

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More U-Substitution: The “Double-U” Substitution with ArcTan(u)

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More U-Substitution:The “Double-U”Substitution with ArcTan(u)

Chapter 5.5

February 13, 2007

- Use Graph & Area ( )
- Use Basic Integral Formulas
- Simplify if possible (multiply out, separate fractions…)
- Use U-Substitution…..

- 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

Extra “x”

Extra “x”

Still have an extra “x” that can’t be related to the substitution.

U-substitution cannot be used for this integral

Returning to the original variable “t”:

Returning to the original variable “t”:

We have the formula:

Factor out the 9 in the expression 9 + t2:

Factor out the a2 in the expression a2 + t2:

We now have the formula:

Returning to the original variable “t”:

It’s necessary to know both forms:

t2 - 2t +26 and 25 + (t-1)2

t2 - 2t +26 = (t2 - 2t + 1) + 25 = (t-1)2+ 25

- Comes from

- 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.