# M119 Notes, Lecture 2 - Get Now DOC

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```					                                        M119 Notes, Lecture 20
7.1 Basic anti-derivatives

f (x)               F (x)
4                    4x + C
m                    mx + b
3x 2                 x3  C
xn                      1
x n 1  C , x  1 ,
(n  1)
1                    ln x  C
x
ex                   ex  C

Ex. f ( x)  4e x  2 x10  5 , find an antiderivative of f(x).

If F(0) = 2, find F(x)

Indefinite Integral
3
 x dx 

          7x  3 


9x 
x3 
dx 
7.2 Integration by u-substitution

e        dx 
5x

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:
u  5x
du
5
dx
du  5dx
1
du  dx
5

We are solving for dx so we can transform dx to du. Note in this case it is reminding us
1
that f(5x) means that the x values are compressed by 5 (i.e times )
5

1  1
e dx   e u  du    e u du
5x

5  5
Important: integrate before putting g(x) back in for u.

Ex.  t  3t 2  7  dt 
3
ex
Ex.      5e x  2
dx 

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:
b

 g ' ( x)dx  g (b)  g (a) , but if the function inside the derivative doesn’t have a prime, we
a
need to make a new function, F(x):
b

 f ( x)dx  F (b)  F (a)
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?

5
x
Ex.    3x
1
2
5
dx
4 ln q  3
10
Ex.   
1
q
dq

2

 x e dx
2 x 1  3

Ex.
2

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