Part 4 (Integration)
Integrate the following: (Sections 4.8 and 5.5)
1) ∫
5𝑥+3
2)
 1dt
5)
√𝑥
 t
1
4)
𝑑𝑥
2
1
2
2
 (sec x  csc x)dx

0
7) 2  x
2
x 3
2
e
x  1dx
8)
1
5
9) ∫ (2𝑠𝑖𝑛𝑥 − 𝑥 + 𝑥 − 4
6
√𝑥 3
2x  3
 2 x 2  6 x  9 dx
𝑙𝑛√𝑥
15) ∫
𝑑𝑥
𝑥
) 𝑑𝑥 10)

18) ∫ 𝑥 𝑠𝑒𝑐(𝑥 )𝑑𝑥
x 2  x  24
9)

 1  cos

4
2
dx
x
2 x csc 2 2 xdx
5 x  16
dx
x2  9
x
dx
d
4 sin t dt
11)
dx 2

13)
sin 3 x cos xdx
0
14)
ex
 4  9e2 x dx
3
𝑥
16) ∫ 3
𝑑𝑥
√𝑥 2 − 1
17) ∫ 𝑡𝑎𝑛2𝑥𝑑𝑥
2
3
2
10 x  5
 cot
6)
dx
ln x
dx
x

e
3
12)
2x
sin 3 x
3)
3
𝑒 ⁄𝑥
19) ∫ 2 𝑑𝑥
𝑥
20) ∫ 𝑒 𝑥 √1 − 𝑒 𝑥 𝑑𝑥
1
21) ∫
𝑎𝑟𝑐𝑠𝑖𝑛𝑥𝑑𝑥
√1 − 𝑥 2
√3⁄
2
22) ∫
0
1
𝑑𝑥
1 + 4𝑥 2
23) Solve the differential equation; f ' ' ( x)  x 2 , f ' (0)  6 , f (0)  3
24) Solve the
35) Bill throws a stone upward from the ground. The stone reaches a maximum
height of 225 ft. What was its initial velocity?
36) An object has a constant acceleration of -42 feet per second squared, an initial
velocity of 5 feet per second, and an initial position of 9 feet. Find the position
function describing the motion of this object.
37) Approximate the area below the graph of f ( x)  4  x 2 and above the x-axis
between x=0 and x=2. Use 4 rectangles and find both the right and left
approximation. (Section 5.2)
38)
Use the limit definition of area to find the actual area below f ( x)  4  x 2 and
above the x-axis between x =0 and x=2. (5.2)
39)
Use the limit process to find the area of the region between the graph of the
function f ( x)  2 x  x 2 and the x-axis over the interval [0, 2].
2
40) Evaluate the definite integral by the limit definition
 (2 x  5)dx
(Section 5.3)
1
51) Find the average value of f ( x)  2 x 2  3 on the interval [2,4]
Think About it
53) Create a function that has a vertical asymptote at x=5 and a horizontal asymptote at y=2
54) Sketch a graph of a differentiable function f that satisfies the following conditions:
f ' ( x)  0 when x<2 f ' ( x)  0 when x>2
lim f ( x)  lim f ( x)  6
x 
x  
55) Sketch the graph of a function that does NOT have a point of inflection at (c, f(c))even though f’’(c)=0
56) True or false; The graph of every cubic polynomial has precisely one point of inflection
61)True or false. If f ' ' (2)  0 , then the graph of f must have a point of inflection at x=2