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
**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
**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
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 . 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 , nothing will change except in step 5 you will have to
replace
*a*and*b*also, that isIn 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:

**
**

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Contact us

Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA

users online during the last hour