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More U-Substitution: The “Double-U” Substitution with ArcTan(u)
Chapter 5.5 February 13, 2007
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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…..
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Evaluate:
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Compare the two Integrals:
Extra “x”
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Compare:
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Evaluate:
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Evaluate:
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Evaluate: We have the formula:
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In general:
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Evaluate:
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But: 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
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Completing the Square:
Comes from
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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.
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Evaluate:
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Try these:
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In groups of two/three, use u-substitution to complete:
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