5.b – The Substitution Rule

f  g  x   g   x  dx
1
Example – Optional for Pattern Learners
Use WolframAlpha.com to evaluate the following.
 e  6 x dx.
2. Evaluate  sin  4 x  12 x
3 x2
1. Evaluate
3
3. Evaluate

2
WolframAlpha
Notation
dx.
integral[f(x),x]
cos  tan x   sec x dx.
2
Notice that each of these are of the form  f  u   u dx,
where u is some function of x. If the antiderivative of f
is F, what will be the answer to the indefinite integral of
this form?
2
The Substitution Rule – The Idea
Evaluate 
How did you arrive at this answer?
cos(e x )  e x dx without using WolframAlpha.
3
The Substitution Rule – The Idea
Let’s use substitution to evaluate

cos(e x )  e x dx
Let u = ex and use this to complete the blanks below.
f  u   _____________
du
 _________________ or du  _______  __________
dx


cos  e x   e x dx   _________ ____ (in terms of u)

cos  e x   e x dx  ____________  c (in terms of x)
_________ ____  __________  c (in terms of u)
4
The Substitution Rule
If u = g(x) is a differentiable function whose range is an interval I
and f is continuous on I, then
 f  g  x  g  x  dx   f u  du
u
Since,
du
u  g  x
du
 g  x
dx
du  g   x  dx
In general, u will
be the inside of
the composition,
but this isn’t
always the case.
5
The Substitution Rule – General Technique
1. (Optional) Rearrange the function so that anything not in the
composition is in front of the dx.
2. Let u be g(x) in the composition. More generally, let u be the
part of the integrand such that du/dx is the other part of the
integrand (with the possible exception of the coefficient of
du/dx – it can be different).
3. Determine du/dx and multiply both sides by dx.
4. Divide both sides by the coefficient, if necessary.
5. (Only Applies to Some Integrals) If there are extra x’s
remaining, solve u for x and substitute for the remaining x’s.
6. Perform the substitution, evaluate the integral, then perform
the back substitution to get it in terms of the original variable.
6
Examples - Evaluate
1.
2.



4. 
3.
5.

3 x  x  5  dx
2
x
y
3
9
3
x
2
 1
2
dx
2 y  1 dy
4
6.
7.
8.
sec 2 tan 2 d
9.
tan 1 x
dx
2
1 x
10.
x sin 1  x 3 / 2  dx

 1  tan   sec
 e sin t dt
5
2
 d
cos t
sin x
dx
2
1  cos x


tan x dx
7
Substitution Rule Twists - Examples
Sometimes choosing the function for u can be challenging. Always
keep in mind that you want to select a part of the integrand such
that it’s derivative gives you the other part of the integrand.
1.

x
dx
4
1 9x
The following have extra x’s remaining in the integrand after the
u substitution. This is fine. Simply solve u for x and substitute.
2.

3
x3  1 x5 dx
3.

x2
dx
1 x
8