Section 7.1 Integration by Substitution

See if you can figure out what functions would give the following derivatives

Recall the chain rule Now imagine taking the antiderivative of both sides Therefore any we can find the derivative of any function of this form using the method of substitution

The Method of Substitution When integrating something of the form we let u = g(x) or the “inside function” Thenand we get an integral that can be done in terms of u Let’s look at our previous examples using this method which I refer to as u substitution

Now we can always adjust our substitution if we are off by a constant For example, let’s find the following antiderivative

Substitution can also be used for definite integrals

WARNING!!! We may not be able to use substitution if anything other than a constant multiple is missing. (We cannot just “add in” a variable) Example “x 2 dx” is not a constant multiple of ”dw = 4x 3 dx”

A more complex substitution is needed to solve a problem such as