Integration by Substitution

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Integration bySubstitution Separable
Differential Equations

• Section 6.2

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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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Properties of Indefinite Integrals
We already know the following integrals based on
what we have learned previously about derivatives
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Trig Integrals
We already know the following integrals based on
what we have learned previously about derivatives
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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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Example (Exploration 1 in the book)
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Example 2
Solve for dx.
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Example 3
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Example (Not in book)
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Example 7
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Example 8
We can find new limits, and then we dont have to
substitute back.
We could have substituted back and used the
original limits.
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Example 8
Using the original limits
Wrong! The limits dont match!
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Example
Dont forget to use the new limits.
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Warm-up


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Separable Differential Equations
A separable differential equation can be
expressed as the product of a function of x and a
function of y.
Example
Multiply both sides by dx and divide both sides
by y2 to separate the variables. (Assume y2
is never zero.)
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Separable Differential Equations
A separable differential equation can be
expressed as the product of a function of x and a
function of y.
Example
Combined constants of integration
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Example 9
Separable differential equation
Combined constants of integration
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Example 9
We now have y as an implicit function of x.
We can find y as an explicit function of x by
taking the tangent of both sides.
Notice that we can not factor out the constant C,
because the distributive property does not work
with tangent.
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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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