Questions of first course and answer

Ministry of Higher Education Primary - Studies
and Scientific Research First class stage
Babylon University Allowed time is 3 Hours
College of Science Examine of Mathematics
Chemistry Department 22 / 1 / 2019

Final Examination Questions of first course for Scholar year 2018 – 2019 First attempt
Remake : Answer Five questions . (10 marks for each question )

Q? 1? A.Find the domain of the function f(x)=?(x^2-2x-15 )


B. Evaluate ?_0^2?(3x^2 dx)/?(x^3+1) dx

? Q2? A.Find the value of lim?(x??)??3x/?(?16x?^2+1)?

B. If y=sec^(-1)?3t and x=?(9t^2-1),then find dy/dx

? Q3? A. Find [H^+ ] if pH=8.5 .
B.Solve for x? 3 cosh?x- 2 sinh??x ?=3.

? Q4? Find the value of A. lim?(x?0)??(2 sin?x-sin?2x)/(x-sin?x )? . B.?lim ??(x? -1) (1+cos?x)/(x^2+2x+1)

Q? 5? Find A.??(2x-8)dx/(x^2-6x+13) B.???x ln?x ? dx


Q? 6? Find A. ???x^3 e?^(-x) dx B.??(x+1)/(x^2-5x+6) dx



Lecturer Fouad H. A. Alsharify Prof. Dr. Abbas Jassim Attiah
Instructor Head of department
??? With My Best Wishes ???


Typical answers
Q? 1? A. x^2-2x-15 ?0
x^2-2x-15 =0 ?(x-5)(x+3)=0 ? x=5 ,-3
? (-??,-3] ,[-3,5] and [5,? ?)?
0?[-3,5] ? -15?0 ? False ?
Thus the domain is D=? (-??,-3]?[5,? ?)?


B. ?_0^2?(3x^2 dx)/?(x^3+1) dx=2? ?(x^3+1)?|_0^2 =2(3-1)=4

? Q2? A.=lim?(x??)??(3x/x)/(?(?16x?^2+1)/x)?=lim?(x??)??3/?((?16x?^2+1)/x^2 )?=lim?(x??)??3/?(16+( 1 )/x^2 )?=( 3 )/( 4 )


B. dy/dt=3/(3t?(?9t?^(2 )-1)) , dx/dt=18t/(2?(?9t?^(2 )-1))
dy/dx=dy/dt×dt/dx=1/(t?(?9t?^(2 )-1))×?(t^(2 )-1)/9t=1/?9t?^(2 )

? Q3? A. pH=-log?[H^+ ]
8.5=-log?[H^+ ]
log?[H^+ ]=-8.5
[H^+ ]=?10?^(-8.5 )=?10?^(0.6-8 )=?10?^(0.5 )×?10?^(-9 )
[H^+ ]=3.16×?10?^(-9) M

B. (3e^x+3e^(-x)-2e^x+2e^(-x))/2=3
{e^x+5e^(-x)=6}×e^x
e^2x-6e^x+5=0
(e^x-1)(e^x-5)=0 ? e^x=1 or e^x=5
x=ln?1=0 or x=ln?5=1.609




? Q4? A. lim?(x?0)??(2 sin?x-sin?2x)/(x-sin?x )?=lim?(x?0)??(2 cos?x-2 cos?2x)/(1-cos?x )?=lim?(x?0)??(-2 sin?x+4 sin?2x)/sin?x ?
=lim?(x?0)??(-2 cos?x+8 cos?2x)/cos?x ?=(-2+8)/1=6

B.?lim ??(x? -1) (1+cos?x)/(x^2+2x+1)=?lim ??(x? -1) (-?sin?x)/(2x+2)=?lim ??(x? -1) (-?^2 cos?x)/2=( ?^2)/2

Q? 5? A.??(2x-8)dx/(x^2-6x+13) =???(2x-6-2)/(x^2-6x+13) dx?
=???(2x-6)/(x^2-6x+13) dx?-???2/(x^2-6x+9+4) dx?
=ln?|x^2-6x+13|-???2/((x-3)^2+4) dx?
=ln?|x^2-6x+13|-tan^(-1)??((x-3))/2?+c

B.u=ln?(x) and dv=xdx ?du=dx/x and v=x^2/2
???x ln?x ? dx=(x^2 ln?x)/2-???x^2/2 ×dx/x?=(x^2 ln?x)/2-???x/2 dx?=(x^2 ln?x)/2-x^2/4+c


Q? 6? A. ???x^3 e?^(-x) dx=- e^(-x) (x^3+? 2x?^2+6x+6)+c
e^(-x) I. x^3 D
e^(-x) + x^3
-e^(-x) -? 2x?^2
e^(-x) + 6x
?-e?^(-x) - 6
e^(-x) 0

B.(x+1)/(x-3)(x-2) =A/((x-3) )+B/((x-2) )
x=3 ? A=4 , x=2?B=-3

??(x+1)/(x^2-5x+6) dx=??(4/((x-3) )-3/((x-2) )) dx
=4 ln?|x-3|-3 ln?|x-2|+c