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

Techniques of Integration so far… 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…..

Evaluate:

Compare the two Integrals: Extra “x”

Compare:

Evaluate:

Evaluate:

Evaluate: We have the formula:

In general:

Evaluate:

But: 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: