AP CALC BC
Integration Summary IV
Name
Integrate each of the following:
e tan 3 x
1.
dx =
cos 2 3 x
2.
3csc 2 x
dx =
4 cot x 9
3. cos3 (3x)dx =
4. cot(3 x) dx =
5. (3 cot 3 x)5 csc 2 3 xdx =
6.
sec13x tan13 x
dx =
4sec13 x 9
7. x cos 5 xdx =
By Parts:
u=
AP CALC BC
Integration Summary III
Name
Integrate each of the following:
1. e 2sin x cos xdx =
3sec 2 x
2.
dx =
4 tan x 9
3. sin 3 (3 x) dx =
4. tan(3 x)dx =
5. (3 tan 3x)5 sec 2 3xdx =
6.
sin13 x
dx =
4 cos13 x 9
7. x sin 2 xdx =
By Parts:
u=
dv=
8.
9.
1
AP CALC BC
Overview of Integrals II
Integrate each completely:
1. esin x cos xdx =
2.
e3 x
dx =
3e3 x 9
3. (3 3cot 2 x) 3 csc 2 2 xdx =
4.
( x 9) 4
dx =
x
5. (3 x + 4) cos 2 xdx =
6. 3cot 2 5 xdx =
Name
7. sin 7 xdx =
8. tan 3 2 xdx =
9. cos5 2 xdx =
10.
AP CALC BC
Integration Summary I
Name
Integrate each of the following:
1. e 2 x dx =
2.
3x
dx =
4x2 9
3. sin(3 x)dx =
4. tan 2 (3 x)dx =
5. (3 tan 2 x) 4 sec 2 2 xdx =
6.
sin 3 x
dx =
4 cos 3 x 9
7. xe x dx =
By Parts:
u=
dv=
8. 3 4 x 5dx =
9.
3x
4
4x2 9
AP CALC BC
Name
Optimization Problems III
Recall: When you are solving an optimization situation, you need to use a diagram,
use information in the problem that is considered auxiliary information and to set
up an equation in two variables, one variable r
AP CALC BC
Optimization II
Name
1. Given the equation y = 9 x 2 , find the area of the largest inscribed rectangle.
2. Given the equation y = 12 x 2 , find the area of the largest inscribed
rectangle.
3. Given the equation y = 25 x 2 , find the area of th
AP CALC BC
Optimization I
Name
1. You have 2300 feet of fence and wish to make a rectangular region of
maximum area. What are the dimensions of such a region?
2. You have 3000 feet of fence and wish to make a rectangular region with a
fence length down th
AP CALC BC
Integration with Trig Identities
Name
Integration XII
Sometimes we are given an integral that we can not integrate unless we use some
trigonometric identities from Pre-Calculus. Below, you will find the list of identities
that we will be using
AP CALC BC
Name
Integration using ln definition for integration Integration X
Note:
n
au du =
au n +1
+ c, n 1
(n + 1)
du
= ln u + c, n = 1
u
Example:
xdx
1
dx = ln 2 x 2 7 + c : letu = 2 x 2 7
2
7)
4
cos x
4 + sin x dx = ln 4 + sin x + c : letu = 4 +
AP CALC BC
Volume by Rotations
Name
Notes: A plane figure that can be sketched using the given equations has a bounded
region. This region is then rotated about a vertical or horizontal line to create a 3-D
figure which we are to find the volume of. Metho
AP CALC BC
Name
Volume by Rotations Worksheet II
Notes: A plane figure that can be sketched using the given equations has a bounded
region. This region is then rotated about a vertical or horizontal line to create a 3-D
figure which we are to find the vol