King Saud University
First semester 1434-1435
106 Math
Final Exam
==============================================================================
Student name:-
student number:-
Section number:-
teacher name:-
Question
Mark
1
2
3
Question
i
ii
iii
Answer
Q1. Choose the correct answer:-
4
iv
5
v
6
vi
vii
7
iix
10
i)
 2i
equal to:
i 1
a) 55
ii)
b) 220
c) 110
d) none of these
b) 3 2 ln 
c) 0
d) none of these
d 3

dx
a)  3 ln 
4

x  1
iii) The function F  x   ln
x  13
a)
x 1
x2 1
2
iv) If
x
 ln e dx 
1
a)
3
2
b)
x7
x2 1
3
, then
2
b) 
3
2
is an antiderivative of the function f x  
c)
7x  1
x2 1
d) none of these
1
e
ln x
dx is equal to:
2
c) 0
d) none of these
Total
ix
x
3
v) Anumber C that that satisfies the M.V.T for the
  3x
2
 1 dx  24 is
1
13
3
a)
b) 
13
3
c) 12
d) none of these
vi) The polar equation corresponding to the rectangular equation
y x 2  y 2  5x ,
a) r  5 cot 
x  o, y  0 is:
b) r  5 tan 
c) r  5 cos
d) none of these
vii) The rectangular equation corresponding to the polar equation  
c) x  0
b) y  0
a) x  y

2
is
d) none of these
iix) A polar coordinate representation of the rectangular point (0,-1) is


a)   1,
3 

2 
 

 2
b) 1,


c)   1,




2
ix) Another polar representation of the point  4,


a)   4,
7 

3 


b)   4,
4 

3 
d) none of these

 is
3


c)  4,
7 

3 
d) none of these
x) A parametric equation of a circle centered at (1,-3) and of radius 1 is
x  1  cos t
a) y  3  sin t
0  t  2
x  1  cos t
b) y  3  sin t
0  t  2
x  1  cos t
c) y  3  sin t
0  t  2
d) none of these
Q2.
i) If Gx   x 
x2

t sin 2 tdt , prove that G x   2 x 2 cos 2 x 2  1  2 x 2
1
ii) a. Prove that
m
 x ln x  dx 
n
x m1
ln x n  n  x m ln x n1 dx
m 1
m 1
b. Use part (a) to find
 x ln x dx


3
c. Use cosh 1 x  ln x  x 2  1 , x  1 find
d
cosh 1 3 x 
dx
Q3.a) Determine whether the following integrals converge or diverge.
2
i)
3
x
2
dx
2
0
ii)
 xe dx
x

b) Evaluate the following integral

sin
x
1
x dx
2
Q4.a) Find the arc length of the curve determined by f x  
x

0
ln 2
b) Evaluate the integral
e2x
 sinh x  cosh x
2
 
cos 2t dt , 0, 
 4
dx
0
Q5.a) Find the area of the region bounded by the curves y  e x
b) Evaluate the integral

4  x2
dx
x
.
,
y  ex
, 0  x 1
Q6.a) Find the volume of the solid formed by revolving the region bounded by the curves
y
x
,
y  x  2 ,
y  0 . (Do not integrate)
i) about x- axis
ii) about y- axis
b) Evaluate the integral
x
2
x
dx
 2x  5
Q7. a) Sketch the region of the polar equation r  sin 2 , then find the area of one leaf.
b) Evaluate the integral  sec 3 x tan 3 xdx .