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

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

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

Evaluate:

Compare the two Integrals: Extra “x”

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

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

Evaluate: Returning to the original variable “t”:

Evaluate: Returning to the original variable “t”:

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

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

Evaluate: Returning to the original variable “t”:

Use: 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

Completing the Square: Comes from

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.

Evaluate:

Try these:

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