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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
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