Integration by Substitution

Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients. Sometimes we have a choice of method.

We have already met this type of integral. See if you can do it. Solution: Reversing the chain rule gives We’ll use this example to illustrate integration by substitution but if you got it right you can continue to use the reverse chain rule ( also called inspection ).

e.g. 1 Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . Find dx by treating like a fraction

e.g. 1 Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx

e.g. 1 Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx

e.g. 1 Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx Integrate: Replace u:

e.g. 1 Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx Integrate: Replace u:

e.g. 2 This is a product but we can’t use integration by parts. Why not? ANS: is a compound function with inner function non-linear. We can’t integrate it. If we chose to integrate x instead, at the next stage we would have a more complicated integral than the one we started with. We substitute as before, but using the inner function of the 2nd factor in the product.

Cancel the extra x e.g. 2 Define u as the inner function: Let Differentiate: Substitute for the inner function and dx Cancel the extra x If x won’t cancel we will have to make an extra substitution. We’ll do an example later.

So, where Integrate: Substitute back:

SUMMARY Substitution can be used for a variety of integrals e.gs. Method: Define u as the inner function Differentiate the substitution expression and rearrange to find dx Substitute for the inner function and dx If there’s an extra x, cancel it If x won’t cancel, rearrange the substitution expression to find x and substitute for it Integrate Substitute back

Definite integration We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u. A definite integral gives a value so we never return to x.

e.g. 1 Let Limits: So,

So,

e.g. 2 Let Limits: So,

So, You will often see this written as where We leave answers in the exact form.

Exercises 2. 1. Give exact answers.

1. Solutions: Let Limits: So

2. Let Limits: So, We can use the log laws to simplify this.