# Integration by Ilg351

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```									    AP Calculus

Integration
U-Substitution & Properties

Laura Habberfield   Per 4
When you take derivatives, you frequently have to use the
chain rule to differentiate. The integration equivalent of the
chain rule is called u-substitution.

 x( x 2  1)5 dx
( x 2  1)
Set up a u = _______       Find
du
2x
dx = _______.
2 x dx
Solve for du = _______
You need to manufacture your du in the original expression.
So you will have to multiply by _____ on the inside and thus
2
1
multiply by ______ on the outside.
2

Now change everything to u.              u 5du
1 u6
Now integrate in terms of u.              6
Finally, change back to the variable x and add C.
1 ( x 2  1) 6  C
6
http://www.mastermathmentor.com/mmm/calc/default.aspx?page=ABManual
Integral Properties
a
   a
f ( x ) dx                         If we start at a and end at a, there is no area.

b                          a
           f ( x) dx    f ( x) dx
From a to b gives an area. From b to a
gives the negative of this area.
a                          b

b                         c               c                       The area from a to b plus the area


a
f ( x) dx   f ( x) dx   f ( x) dx
b               a
from b to c = the area from a to c
when f(x) is continuous on the
integral [a,c]

http://www.mastermathmentor.com/mmm/calc/default.aspx?page=ABManual
Integral Properties (continued)

#                   #                If the integral is symmetric in the y-axis, its limits
2  f ( x) dx   f ( x) dx              can be from 0 to a # and multiplied by 2, or from
0                   #                   the negative number to the positive number.

   f ( x) dx  0           If the integral is symmetric in the origin, it will always = 0.

   f ( x) dx  2   f ( x) dx   2               The integral of two functions or lines is
equal to the integral of each piece alone.
Using integrals numerically is generally done with
Riemann Sums or the Trapezoid Rule, which are both
covered under another review section.

Try these multiple choice questions on what you just reviewed! Answers follow.

1. If            ,            and               , find the value of
A. 3
B. -3
C. 2
D. -2
E. Can’t be determined

http://webs.bcp.org/sites/jmolina/summer/Indefinite%20a
nd%20definite%20integrals,%20and%20u-substitution.htm
2. Find the value of the integral

A.

B.

C.

D.

E.



         e2cos x sin x dx 
2
3.
0
A.
e2  e
B.     1
C.
e2
D.     0

E.    Does not exist
Calculus AB/BC
http://webs.bcp.org/sites/jmolina/summer/Indefinite%20a
nd%20definite%20integrals,%20and%20u-substitution.htm       Maxine Lifshitz
4. Let f(x) be an odd function and g(x) be even.           A. II
Which of the following statements are true?             B. III
2
I.       2
f ( x) dx  0                    C. I and II
II.                                                D. I and III
2
2
g ( x) dx  0
2
III.      
2
f ( x) g ( x) dx  0             E. I, II, and III

5. Find
          
x 2 x 3  1 dx        
5

( x 3  1) 6                             1. A
C
A.        18                                  2. C
( x 3  1) 6
B.                C                          3. A
6
C.   ( x 3  1) 6                             4. D
C
2                                  5. A
D.   6( x 3  1) 6  C
E.   2 x( x 3  1)5  C                                      Calculus AB/BC
Maxine Lifshitz
1. Traffic flow is determined as the rate at which cars pass through an
intersection, measured in cars per minute. The traffic flow at a
particular interesection is modeled by the function F defined by:

t
F (t )  82  4 sin   for 0  t  30
2
Set up an integral, but do not solve, to determine how many cars pass
through the intersection over the 30-minute period.

30              t
0
82  4 sin  
2

http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html?calcab
2. At an intersection in Thomasville, Oregon, cars turn left at the rate
t
L(t )   60 t sin         2

3
cars per hour over the time interval 0<t<18 hours. The graph of
y=L(t) is shown below. To the nearest whole number, find the total
number of cars turning left at the intersection over the time interval
0<t<18 hours. Calculator available.

18              t
       60 t sin   dt  1658 cars
2
0
3

http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html?calcab

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