Section 4.5

Integration by Substitution

Chain rule for integrationRemember chain rule:

Now we need to go backwards and take the antiderivative:

Or

)())(())(( xgxgFxgFdx

d

CuFCxgFdxxgxgF )())(()())((

CuFdxdx

duuf )()(

Ex1a: dxxx42 12

Ex1b: dxxx 13 32

Ex1c: dxxx 3tansec2

Sometimes we don’t have the exact derivative of u in the integral

Ex2a: dxxx62 1

Ex2b: dxx 12

Ex2c:

dxx

x

54

7

3

2

Ex3a: xdx2sin

Ex3b: dx4

csc2

Ex4: dxxx )3cos3(sin2

Ex5a: dxxx 12

Ex5b:

dx

x

x

4

12

What about definite integration?

Ex6a:

1

0

32 1 dxxx

Ex6b:

5

1 12dx

x

x

AP Question

Which of the following is an equivalent integral expression for

3

2

32 2 dxxx