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Integral
calculus |
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The
indefinite integral |
Substitution rule |
Evaluating the indefinite integrals using substitution rule, examples
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Evaluating the indefinite integrals using substitution rule,
solutions
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Substitution
rule |
Let
f
and g
are differentiable, with range of g
contained in the domain of f,
and let F
be antiderivative of f
such
that F'
(u)
= f (u) where u
= g (x), then |
![](SR.gif) |
by
applying the chain rule |
F(g (x))' =
F' (g (x)) g' (x) = f
(g
(x)) g' (x) . |
Thus,
obtained is substitution rule |
![](SR1.gif) |
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Evaluating
indefinite integrals using substitution rule, examples
|
Evaluate
the following indefinite integrals using substitution rule |
21. |
![](SE21.gif) |
22. |
![](SE22.gif) |
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23. |
![](SE23.gif) |
24. |
![](SE24.gif) |
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25. |
![](SE25.gif) |
26. |
![](SE26.gif) |
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27. |
![](SE27.gif) |
28. |
![](SE28.gif) |
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29. |
![](SE29.gif) |
30. |
![](SE30.gif) |
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Evaluating the indefinite integrals using substitution rule,
solutions
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Example:
21.
Evaluate |
![](SE21.gif) |
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Example:
22.
Evaluate |
![](SE22.gif) |
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Example:
23.
Evaluate |
![](SE23.gif) |
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Example:
24.
Evaluate |
![](SE24.gif) |
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Example:
25.
Evaluate |
![](SE25.gif) |
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Example:
26.
Evaluate |
![](SE26.gif) |
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Example:
27.
Evaluate |
![](SE27.gif) |
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Example:
28.
Evaluate |
![](SE28.gif) |
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Example:
29.
Evaluate |
![](SE29.gif) |
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This
integral can also be solved using decomposition of rational function
into a sum of partial fractions.
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Example:
30.
Evaluate |
![](SE30.gif) |
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