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

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….

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

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  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: Extra “x”

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

  14. Completing the Square: • Comes from

  15. 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:

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