Calculus
u-substitution guided notes
NAME________________________
Remember the chain rule?
d
 f ( g ( x) )  = __________________
dx 
and
d n
u  = __________________
dx  
And, remember integration?
Since
d
[sin( x)] = ___________ , then we know that ∫ __________ dx = sin( x) + C
dx
So check these out:
(A) Simplify
Since
d 
4
4
x + 3)  = ___________ , then ∫ __________ dx = ( x + 3 ) + C
(

dx 
(B) Simplify
Since
10
d  2

x
+
5
(
)

dx 
10
d  2
2
 = ___________
x
+
5
__________
dx
=
x
+5
(
)
,
then
∫

dx 
(
(C) Simplify
Since
d 
4
( x + 3) 

dx
d
u n 
dx
d n
u  = ___________ , then ∫ __________ dx = u n + C
dx
Or more appropriately written as
∫ __________ du = __________ + C
This is our formula for the reverse chain rule, I.e., u-substitution
)
10
+C
What is a key indicator when differentiating that you will use chain Rule?
What do you think is a key indicator when integrating that you will use u-substitution?
Let’s walk through our process….
∫ ( 2 x + 5) dx
3
Original Integral
u = _______
du
dx
Define u
= _______
Differentiate u
du = _______
____ = _______
Solve for du
Rearrange du equation
∫ _______________ du
___ ∫ _______________ du
Simplify u-integral
_______________ + C
__________________
Integrate with respect to u
Substitute to get in terms of x
Rewrite integral with respect to u
Now try these:
1)
∫ 2x ( x
2
+ 9 ) dx
5
Why does u-substitution exist?
2)
∫ x (x
3
4
− 1) dx
2