New Mexico Tech

Solutions Practice Questions for Exam 4
4 1
1 4 4 3
 3
2
1
1.
  2 x  4 x  2   dx  x  x  4 x  ln x  C
x
x
2
3


2
3
3
6
 x x x
2.
dx   x5/3  x2/3  x1/6 dx  x8/3  x5/3  x7/6  C

3
8
5
7
x

1
1
3.
 sin 2x  cos3x  dx   2 cos  2x   3 sin 3x   C
2
4.
 sec   sec tan d  tan  sec  C

5.
6.
7.
8.
9.
10.
11.

 e


1
1
 e3 x  dx  e2 x  e3 x  C
2
3
x
1
1
4
1
1
 x
 1 dx  1 
dx  
du  arctan    C , let u  , du  dx



2
2
2
 16  x
4
4
16  1  x /16
16  1  u
4
4
3
3 1
3

dx

du

arcsin  2 x   C
Let u  2 x , du  2dx , 

2  1 u2
2
 1  4 x2
2x
Let u  3x2 , du  6 xdx 12xe3x dx  2 e3x  6x  dx  2 eu du  2eu  C  2e3x  C
2
2


x 1
4
12.
13.
Let u  sin x , du  cos xdx so  sin10 x cos x dx   u10du 
15.
16.
2
2
2
3/2
Let u  1  cos x , du  sin xdx so  sin x 1  cos xdx   udu  u 3/2  C  1  cos x   C
3
3
2
x
1
2
Let u  x2  4 , du  2xdx so 
dx  
 2
 du  ln u  C  ln x  4  C
 x 4
u
 3
1
 2 dx  3cosh x  C
 x 1

1
Let u  x  1 . du 
dx so 

2 x

14.
Math 131
2 x

1 11
1
u  C  sin11 x  C
11
11
1
1
1
Let u  x3 , du  3x2dx so  x2 sec2  x3  dx   sec2 udu  tan u  C  tan  x3   C
3
3
3
1
1

dx  ln 2 x  5  C

 2x  5
2
2
Let u  x 1, du  2xdx , if x  0 then u  1 , if x  3 , then u  4
4
1
1  2  3/2 4 1 3/2
7
2
0 x x 1 dx  2  u du  2  3  u 1  3  4 1  3
3
2
17.
e
2
19.
1
2
4 x8
1
1
dx  e4 x8   e16  1
4
4
2
1   3 
 1 2
sin    sin  0   



  4 
   2 
0
0
1
1
1
dx  
Let u  ln x , du  dx , so 

 du  ln u  C  ln ln x  C
 x ln x
u
x
3/4
18.

5
1
1
dx   u 4du  u5  C  1  x  C
5
5
 cos  x  dx 
1
sin  x 
3/4

1
1
1
6
  arctan x 
dx   u5du  u6  C   arctan x   C
dx , so 
Let u  arctan x , du 
2
2
6
6
1 x
 1 x
5
20.
8
21.
0
6
2
8
f  x  dx   f  x  dx   f  x  dx 

0
2
22.
 h "( x)dx  h '  x 
2
1
0

2
8
8
0
6
f  x  dx   f  x  dx   f  x  dx 
6
 f  x  dx .
2
 h '  2   h ' 1  5  2  3
1
4 x2
9
23.
1
2
1
1
F  x  
 dt   
 dt , find F '  x   2 8x    .
2 t
t
4x
x
9
4x
24.
x
1
d arctan t
e
dt  earctan x

dx 0
a.
d arctan x
e
dx  0
b.
dx 0
c.
1
1
d arctan x

dx  earctan x 0  earctan1  earctan 0  e /4  1
 e

 dx
0
3
25.
Estimate
x
2
 2  dx by using a Riemann sum with n  4 and the right end points.
1
3
26.
1/2
1  x
1 2
1
dx

x

1




3  0  x2  1
3
3
0
3


10  1
0
4
27.
4
28.
3

4
2
2
f ( x)dx  10 , let u  2 x , du  2dx ,
0
2

0
f  2 x  dx 
4
1
1
f  u  du  10  5 .

20
2
3
29.
 f ( x)dx  2  1  1  2  1  1
2
30.
4
1 3

1 3

 28  10  38
1 x  9 dx  1  x  9 dx  3  x  9 dx    3 x  9 x 1   3 x  9x 3    3    3   3
3
2
Find the area between f  x   x 2 , g  x   x  2 for x  2 to x  2 .
31.
Find the area between the curve f  x   x 2  3x  4 and x-axis between x  1 and x  4 .
32.
Find the area bounded by the curves y  2  x , y  x , and x  0 .
Note the questions above are simply a sample of questions for the exam; it is possible that other types of
questions may appear on your exam.