MCQs from Unit-2

∞ −x
1. ∫ e√ x dx is an improper integral of
0

(a) First kind (b) Second kind (c) Third kind (d) None of these
1 x
2. ∫ ex 2 dx is an example of improper integral of
0

(a) First kind (b) Second kind (c) Third kind (d) None of these
∞ 2
3. ∫ cosx2 x dx will be
1

(a) Convergent (b) Divergent (c) Oscillate (d) None of these

∞

4. ∫ ( x−2 1) ( x−3 ) dx is
4
1
(a) Divergent (b) Converge to log 2 (c) Converge to 3 log 2 (d) Converge to log
2
∞

5. ∫ x e−x dx
2
2 −2 3
(a) Converges to 2 (b) Converges to (c) Converges to (d) Converges to
e e2 e2
−3
e2
3
1
6. ∫ 3
dx
−3 ( x+2 )
13 −13 12
(a) Converges to (b) Converges to (c) Divergent (d) Converges to
25 25 25
1

7. The improper integral ∫ x log x dx

0
1 −1
(a) Converges to (b) Converges to (c) Diverges to ∞ (d) Diverges to −∞
4 4
∞
1
8. ∫ 2
dx is
e x ( log x )
(a) Convergent (b) Divergent (c) Oscillate (d) None of these
1
1
9. The integral ∫ dx is improper due to integrand is not defined for
−1 1+ x
(a) x=1 (b) x=−1 (c) x=0 (d) None of these
∞
1
10. The integral ∫ dx will be convergent if
a xp
(a) p=1 (b) p>1 (c) p<1 (d) None of these
11. Which is Eulerian integral of second kind?
∞ ∞
−x n−1 −x n−1
(a) ∫ e x dx , n< 0 (c) ∫ e x dx, n> 0
0 0
1 1
m−1
(b) ∫ x ( 1−x )n−1 dx , m<0 , n<0 (d) ∫ x
m−1
( 1−x )n−1 dx , m>0 , n>0
0 0
12. Which is Eulerian integral of first kind?
∞ ∞
−x n−1 −x n−1
(a) ∫ e x dx, n< 0 (c) ∫ e x dx, n> 0
0 0
1 1
m−1
(b) ∫ x ( 1−x )n−1 dx , m<0 , n<0 (d) ∫ x
m−1
( 1−x )n−1 dx , m>0 , n>0
0 0
∞
x n−1 dx is
−kx
13. The value of ∫ e
0
Γn Γ ( n+1 ) Γn
(a) n (b) (c) (d)
k kn k n +1
Γ ( k +1 )
k n+1
∞ 1
n
−x
14. The value of ∫ e dx is
0

(a) Γ n (b) n Γ n (c) ( n+1 ) Γ n (d) n Γ ( n+1 )

15. Γ ( n+1 ) =n ! can be used when
(a) n is any integer (b) n is positive integer (c) n is negative integer (d) n is any real number

1
8
16. What is value of ∫ ( log x ) dx ?
0

(a) 5 ! (b) 6 ! (c) 7 ! (d) 8 !

∞
1
17. What is the value of ∫ x
dx ?
0 c
1 2 π 1
(a) (b) (c) (d)
log c log c log c 2 log c
1
dx
18. The value of ∫ is
0 √−log x
(a) −√ π (b) √ π (c) π √ 2 (d) −π √2
19. The value of Γ ( −52 ) is
8 −15 −8 15
(a) √π √π (b)
(c) √π (d) √π
15 8 15 8
1
20. What is the value of Γ m Γ m+ , where m>0 ?
2 ( )
√π Γ 2m √2 π Γ m √π Γ 2 m √2 π Γ m
(a) 2 m +1 (b) 2m −1 (c) 2m −1 (d)
2 2 2 22 m +1
1
( )
21. The value of β , 2 will be
2
3 4 3 2
(a) (b) (c) (d)
4 3 2 3
π
2
22. ∫ sin 2 x cos2 x dx will be equal to
0
π π π 32
(a) (b) (c) (d)
8 16 32 π
23. The value of Γ ( 14 ) Γ ( 34 ) is
π 2
(a) π √2 (b) (c) √ (d) 2 √ π
√2 π
π
2
24. ∫ sin 3 x dx will be equal to
0
2 3 1 1
(a) (b) (c) (d)
3 2 3 2
25. What is the equation for the volume enclosed by revolving f ( x ) around the x axis between x=a
and x=b ?
b b b b
2 ' 2 2 ' 2
(a) ∫ π [ f ( x ) ] dx (b) ∫ π [ f ( x ) ] dx (c) ∫ [ f ( x ) ] dx (d) ∫ [ f ( x ) ] dx
a a a a
2 3
26. What is the volume made by revolving y=x −x around the x axis between x=0 and x=1 ?
π 1 1 π
(a) (b) (c) (d)
105 12 105 12
27. What is the volume generated when the region surrounded by y= √ x , y=2 and y=0 is revolved
about y−axis ?
32 32 π 5π
(a) 32 π (b) (c) (d)
5 5 32
28. What is the equation of surface if the curve y=f (x ), a ≤ x ≤ b rotated about x−axis ?
b b
2 2
(a) ∫ 2 πy √ 1+ { f (x) } dx
a
'
√
(c) ∫ 2 πy 1+ { f ( x) } dx
a
b b
2 2
√
'
(b) ∫ 2 π 1+ { f (x) } dx
a
(d) ∫ 2 π √1+ { f ( x) } dx
a

29. What is the surface area of the solid when the line y=x , 1≤ x ≤ 2 is revolved around x−axis ?
(a) 2 √ 3 π (b) 3 √ 2 π (c) √ 3 π (d) √ 2 π
30. What is the surface area of the solid when y=x , 0 ≤ y ≤ 1 is revolved around y−axis ?
(a) 2 √ 3 π (b) 3 √ 2 π (c) √ 3 π (d) √ 2 π

Answers
1. (c) Third kind 2. (b) Second kind 3. (b) Divergent 4. (b) Converge to log 2
3 −1
5. (c) Converges to 6. (c) Divergent 7. (b) Converges to 8. (a) Convergent
e2 4
∞
−x n−1
9. (b) x=−1 10. (b) p>1 11. (c) ∫ e x dx, n> 0
0
1
m−1 Γn
12. (d) ∫ x ( 1−x )n−1 dx , m>0 , n>0 13. (a) 14. (b) n Γ n
0 kn
1
15. (b) n is positive integer 16. (d) 8 ! 17. (a) 18. (b) √ π
log c
−8 √π Γ 2 m 4 π
19. (c) √π 20. (c) 2m −1 21. (b) 22. (b)
15 2 3 16
b
2 2 π
23. (a) π √ 2 24. (a) 25. (a) ∫ π [ f ( x ) ] dx 26. (a)
3 a
105
 b
32 π 2
27. (c)
5 a
√ '
28. (c) ∫ 2 πy 1+ { f ( x) } dx 29. (b) 3 √ 2 π 30. √ 2 π