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.

Write down the given integral


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


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.
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.


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.
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.
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


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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