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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