Integration by Substitution

Recognizing the “Outside-Inside” Pattern

From doing derivatives we need to recognize the integrand above is a composite function from the from the ““derivative of the outside times derivative of the outside times the derivative of the insidethe derivative of the inside”” (chain rule). (chain rule).

Cx 32 1

3

1 “+ C” since this is an indefinite integral

Think of this function as 2 functions: f(x) and g(x)

As a composite function then:

Now look at the original integral:

2xxf 12 xxg

22 1 xxgf

insideoutside

f(g(x)) g’(x)

Read this as “the antiderivative of the outside function with the inside function plugged in…plus C”

Let’s Practice !!!

Let u = x3

du = 3x2

du sin u

Let u = x4 + 2

du = 4 x3

1/4 4

u du

More Practice !!!

Here are some problems for you to work on!!!

Less Apparent Substitution

Let u = x – 1 du = dx

x = u + 1x2 = (u + !)2

(u + 1) u du