Calculus 6.2

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5.3 Integration by Substitution
Greg Kelly Hanford High School Richland,
Washington
M.L.King Jr. Birthplace, Atlanta, GA
Photo by Vickie Kelly, 2002
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The chain rule allows us to differentiate a wide
variety of functions, but we are able to find
antiderivatives for only a limited range of
functions. We can sometimes use substitution to
rewrite functions in a form that we can integrate.
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Example 1
The variable of integration must match the
variable in the expression.
Dont forget to substitute the value for u back
into the problem!
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Guidelines for Making a Change of Variables

• Choose a substitution u g(x). Usually, it is
  best to
• choose the inner part of a composite
  function.

2. Compute du g(x)dx.
3. Rewrite the integral in terms of the variable
u.
4. Evaluate the resulting integral in terms of u

• Replace u by g(x) to obtain antiderivative in
  terms
• of x.

6. Check your answer by differentiating.
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Example
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Example 2
Solve for dx.
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Example 3
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Example
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Example 7
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In another generation or so, we might be able to
use the calculator to find all integrals.
Until then, remember that half the AP exam and
half the nations college professors do not allow
calculators.
You must practice finding integrals by hand until
you are good at it!
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