Techniques of Integration: Substitution

Many integrals are hard to perform at first hand. A smart idea consists in ``cleaning'' them through an algebraic substitution which transforms the given integrals into easier ones. Let us first explain how the substitution technique works.

1
Write down the given integral

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2
Come up with a substitution u = u(x).
3
Ideally you may want to find the inverse function of u(x), meaning that you will find x = x(u).
4
Differentiate to find dx = x'(u) du.
5
Back to the given integral and make the appropriate substitutions

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6
Check after algebraic simplifications that the new integral is easier than the initial one. Otherwise, go back to step 2 and come up with another substitution.
7
Do not forget that the answer to tex2html_wrap_inline29 is a function of x. Therefore once you have finished doing all your calculations, you should substitute back to the initial variable x.

Remarks.

1
In general, if the substitution is good, you may not need to do step 3. Indeed, from u= u(x), differentiate to find du=u'(x)dx. Then substitute the new variable u into the integral tex2html_wrap_inline29 . You should make sure that the old variable x has disappeared from the integral.
2
A better substitution is sometimes hard to find at first hand. Therefore we do not recommend spending a lot of time in step 2 trying to find it. After a while you may start to have a good feeling for the best substitution.
3
If you are given a definite integral tex2html_wrap_inline45 , nothing will change except in step 5 you will have to replace a and b also, that is

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In this case, you will never have to go back to the initial variable x.

The following examples illustrate cases in which you will be required to use the substitution technique:

[Calculus] [Integration By Parts] [Integration of Rational Functions]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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