An important integration rule.
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We now collect some rules for the integration of more complicated
functions. Unfortunately, integration is not as straightforward as
differentiation and one often has to have an “inspired guess” to find
out the antiderivative.
Integration by Substitution
Theorem 1 (Integration by Substitution). Let
be an Interval, be continuous and be continuously
differentiable. Then
Proof: Let be an antiderivative of . Then, according to the chain rule, the
function is
differentiable with
Therefore,
As a direct conclusion of this results, we can formulate the
following:
Theorem 2 (Integration by Substitution II). Let
be an interval, be continuously
differentiable and injective with inverse function . Let with . Then
Example 3. We can use the substitution rule to
determine the area of an ellipse. The equation of an ellipse is given by
This leads to As
a consequence, the area of an ellipse is given by Now we set and . According
to the substitution rule, we now have With
we obtain
Note that integration by substitution can also be applied by using
the following formalism for determining : Consider the
substitution
and “a formal multiplication with yields . For a formal
determination of the integration bounds, we consider the equations , and thus , . Integration by
substitution can then be formally done by
Example 4.
For ,
determine
Consider the “new variable” .
Then and
and
thus . The
integration bounds are given by and and thus
For , , determine Consider the
substitution . Then and thus . For the integration bounds,
consider and which yields , . We now get
For ,
determine
Consider the “new variable” . Then and and
By using the substitution rule, we can also integrate expressions of
type .
For a differentiable function with for all holds
Proof: For , the above integral is of
type
and thus, the result follows by the substitution rule.
Example 5.
For
holds
For holds
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