5.4
Integration by Substitution
Review: (Chain Rule)
[ ( ())] =
0( ()) 0()
From the definition of an antiderivative, it following that Z
0( ()) 0() = ( ()) +
= () +
(a) The Power Model : If = [ ()]where is a differentiable function of and ∈ Q ( 6= 1) then = [ ()] −1 or, equivalently, [ ] = −10
(b) The Trigonometric Model
(1) [sin ] = (cos ) 0 (2) [cos ] = (− sin ) 0 (3) [tan ] = (sec2) 0 (4)
[cot ] = (− csc ) 0
(5) [sec ] = (sec tan ) 0 (6) [csc ] = (− csc cot ) 0 (c) The Exponetial Model : = () where is a differentiable function of
Then
=
()
Theorem 65 Let be a function whose range is an interval , and let be a function that is continuous on . If is differentiable on its domain and is an antiderivative of on then Z ( ()) 0() = ( ()) + If = () then = 0() and Z () = () + 71
Remark 3 ( () + ) = 0()
Theorem 66 If is a differentiable function of then Z [ ()]0() = [ ()] +1 + 1 + 6= 1 Equivalently, if = () then Z = +1 + 1 + 6= 1 Example 132 Evaluate R(2+ 5)7(6)
Example 133 Evaluate R5 cos 5 Example 134 Evaluate R √2− 1 Example 135 Evaluate R4√− 9 Example 136 Evaluate Rsin2 cos Example 137 Evaluate R −4 (1−22)2 Example 138 Evaluate R 3423+2+8 Example 139 Evaluate R √2 4+1 and R6 2 2 √ 4+1 Example 140 Evaluate R2 72
