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. 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

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

Chapter 5.5

February 13, 2007


Techniques of integration so far
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
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




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

Extra “x”


Compare
Compare:

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

U-substitution cannot be used for this integral


Evaluate1
Evaluate:

Returning to the original variable “t”:


Evaluate2
Evaluate:

Returning to the original variable “t”:


Evaluate3
Evaluate:

We have the formula:

Factor out the 9 in the expression 9 + t2:


In general
In general:

Factor out the a2 in the expression a2 + t2:

We now have the formula:


Evaluate4
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



Use to solve
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.


Evaluate5
Evaluate: solve:


Try these
Try these: solve:



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