‫بسم هللا الرحمن الرحيم‬
Kingdom of Saudi Arabia
‫الـمـمـلكـة الـعـربـيـة الـسـعـوديـة‬
‫وزارة الـتـعـلـيـم الـعـالـي‬
‫جـامـعـة الـمـجـمـعـة‬
‫كلية العلوم بالزلفي‬
Ministry of Higher Education
Majmaah University
College Of Sciences in Alzulfi
16/2/1434 -:‫تاريخ االمتحان‬
‫ الثالث‬-:‫المستوى‬
‫ مدخل لحساب التكامل‬-:‫اسم المادة‬
MAT 102 -:‫رقم ورمز المادة‬
Answer the Following Question:
Q1) a) correct the following statements (if it’s incorrect)
𝑥
(any statement mark)
𝑥
1)
∫ sin( 2) 𝑑𝑥 = − cos( 2) + 𝑐
2)
∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝑐
3)
∫1 √5𝑥 + 4 𝑑𝑥 =
4)
∫ 𝑥 2 +5 𝑑𝑥 = − ln|𝑥 2 − 5| − √5 tan−1(√5) + 𝑐
5)
∫ cot(𝑥) 𝑑𝑥 = ln|sin 𝑥| + 𝑐
2
2
9
632
15
𝑥−4
4
𝑥
3
𝑏) Define Improper integral and calculate the following improper integral ∫0
𝑑𝑥
(6 marks)
√9−𝑥 2
3
c) Find the length of the arc of the curve 𝑦 = 𝑥 2 from 𝑥 = 0 to 𝑥 = 5
(7 marks)

 sin x
Q2) a) by using trapezoid rule find the approximate integral
dx
(6 marks)
0
x
b) Calculate i )
2
dx
 a2
c) Calculate by contraction
ii )  sec xdx
x
n
iii )  x ln xdx
iv )  x 5 e x dx
3
(6 marks)
e ax dx
(6 marks)
1
Q3) a) Calculate i ) x 2  1 dx
ii ) 
dx
x 2  2x  1
x3
iii )  3
dx
x x
iv ) 
(2 x  3) 2
1
3
dx
(6marks)
(2 x  3)  1
b) Find the volume generated by revolving about 𝑥-axis the area between the first arch of the cycloid
𝑥 = sin 𝑡, 𝑦 = 1 − cos 𝑡 And the 𝑥-axis
(6 marks)
c) Find the area bounded parabola 𝑦 = 6𝑥 − 𝑥 2 and 𝑦 = 𝑥 2 − 2𝑥
(6 marks)
1
d) By using Simpson’s rule find the absolute error and related error of the integral
x
4
dx
(6 marks)
0
Dr: Mohammed Mohammed khalaf