Substitution
Neil Trivedi
Teacher
Contents
Substitution
Integration by substitution is a method that helps simplify tricky integrals by changing the variable to something easier. We’d take a part of the integral (such as a bracket or complicated expression) and call it something simpler (usually ).
Here are the general steps for substitution for indefinite and definite integrals.
Integration by Substitution (Indefinite Integrals)
1) Choose a substitution: Suppose or .
2) Find or , so we can express in terms of .
3) Rewrite the integrals in terms of and .
4) Integrate with respect to .
5) Use the substitution from step 1 to rewrite the integrated expression back to .
Integration by Substitution (Definite Integrals)
1) Follow steps 1, 2, and 3 above.
2) Change the limits of integration from to using the substitution.
3) Evaluate the definite integral.
Note: The limits have already changed from to , so we don’t need to return to , like we would have to in step 5 when computing indefinite integrals.
Example 1:
Using the substitution , find .
Step 1: Change into .
Step 2: Rewrite the integral in terms of and . Replace the in the bracket with and replace with .
Note: It is very important not to force substitutions. We leave the as it will likely cancel later.
Step 3: Integrate the expression with respect to .
Step 4: Use the substitution to rewrite the integrated expression in terms of . is already the subject of the equation so we can immediately substitute to replace .
Note: This integral could easily have been evaluated by using the reverse chain rule, but this example is used to show how substitution works with a familiar example. All reverse chain rule problems can be completed using substitution which is why you will often see NeilDoesMaths mock mathematicians who integrate simple integrals like using substitution.
No answer provided.
Example 2:
Using the substitution , find .
This is an example where substitution is necessary as this integral cannot be completed using reverse chain rule.
Step 1: Change into .
Step 2: Rewrite the integral in terms of and . We replace with and with .
Simplify the denominator by using the trigonometric identity .
Therefore, we’ll get:
Step 3: Integrate the expression with respect to .
Step 4: Use the substitution to rewrite the integrated expression in terms of . Our substitution is . First, We need to rearrange the substitution to make u the subject.
Substitute to replace in our integrated expression.
No answer provided.
Example 3:
Use a suitable substitution to evaluate .
Step 1: First, we choose a substitution. We’ll use , which implies that .
Note: It is very rare that they will not give us the substitution in normal maths, it is much more common in further maths. However, it’s important to get enough practice so we could recognise suitable substitutions. In this example, we could have also chosen .
Step 2: Find , so we can express in terms of .
Using implicit differentiation, we get
Step 3: Use the substitution to change the limits of integration from to .
Step 4: Rewrite the integral in terms of and . Replace the with , with and the limits of integration.
Unlike example 1, the has not cancelled completely to leave us with an integral independent of . We need to replace using our original substitution.
Substituting into our integral, we notice the and the cancel. So, we get
Note: The is a multiplier so we can take it out of the integral.
Step 5: Integrate the expression with respect to .
Note: As mentioned before, the limits of integration were changed from to , and we evaluated the integral with respect to , so we don’t need to change the integrated expression back to .
No answer provided.
Challenging Questions