M119 Notes, Lecture 20
7.1 Basic anti-derivatives
f (x) F (x)
4 4x + C
m mx + b
3x 2 x3 C
x n 1 C , x 1 ,
1 ln x C
ex ex C
Ex. f ( x) 4e x 2 x10 5 , find an antiderivative of f(x).
If F(0) = 2, find F(x)
7.2 Integration by u-substitution
For integrals containing the composition of functions f(g(x)), one method that might
work is a u-substitution. Let g(x) = u
So for this problem:
We are solving for dx so we can transform dx to du. Note in this case it is reminding us
that f(5x) means that the x values are compressed by 5 (i.e times )
e dx e u du e u du
Important: integrate before putting g(x) back in for u.
Ex. t 3t 2 7 dt
Ex. 5e x 2
7.3 The fundamental theorem
From 5.5, we learned about the fundamental theorem:
In addition to acting as area, the integral can also act as an antiderivative:
g ' ( x)dx g (b) g (a) , but if the function inside the derivative doesn’t have a prime, we
need to make a new function, F(x):
f ( x)dx F (b) F (a)
F ' ( x) f ( x)
If we now combine this concept with the anti-derivative formulas we learned in 7.1 and
7.2, we can now calculate several definite integrals without the aid of our calculator
Find the area between y x 3 and y x 2 2 x by doing the following steps:
A) Draw the two graphs
B) Write out an integral that will find the area.
C) Estimate the value of the integral using your calculator.
Ex. Oil is leaking out a tanker at a rate of r (t ) 50 e 0.02t thousand liters per minute
A) At what rate is the oil leaking out at t=0? At t=60
B) How many liters have leaked out at t=0? After the first hour?
4 ln q 3
x e dx
2 x 1 3