ECAT Mathematics
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ECAT Mathematics
Chapter 17: Integration
80 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
∫x^n dx (n≠−1) =
A
x^{n+1}/(n+1)+C
B
nx^{n-1}+C
C
x^n/n+C
D
nx^{n+1}+C
1
∫1 dx =
A
0
B
1/x
C
x+C
D
lnx+C
2
∫e^x dx =
A
xe^{x-1}
B
e^x+C
C
e^{x+1}
D
xe^x+C
3
∫sinx dx =
A
cosx+C
B
−cosx+C
C
sinx+C
D
−sinx+C
4
∫cosx dx =
A
sinx+C
B
−sinx+C
C
cosx+C
D
−cosx+C
5
∫sec²x dx =
A
secx tanx+C
B
tanx+C
C
cscx+C
D
−cotx+C
6
∫1/x dx =
A
x+C
B
lnx+C
C
ln|x|+C
D
x²/2+C
7
∫a^x dx =
A
a^x+C
B
a^x/lna+C
C
xa^x+C
D
a^x lna+C
8
∫cscx cotx dx =
A
cscx+C
B
−cscx+C
C
cotx+C
D
−cotx+C
9
∫secx tanx dx =
A
secx+C
B
−secx+C
C
tanx+C
D
sec²x+C
10
The definite integral ∫₀¹ x dx =
A
0
B
1/4
C
1/2
D
1
11
∫₀^π sinx dx =
A
0
B
1
C
2
D
π
12
Integration by substitution: ∫f(g(x))g'(x)dx. Let u=g(x), then:
A
du=g'(x)dx
B
du=f(g(x))dx
C
du=g(x)dx
D
du=f(x)dx
13
∫2x(x²+1)^3 dx using u=x²+1:
A
(x²+1)^4/4+C
B
(x²+1)^4+C
C
(x²+1)^3/3+C
D
2(x²+1)^4+C
14
∫sin²x dx = ∫(1−cos2x)/2 dx =
A
x/2−sin2x/4+C
B
x−sinx/2+C
C
−cos²x+C
D
sin²x/2+C
15
∫cos²x dx =
A
x/2+sin2x/4+C
B
x/2−sin2x/4+C
C
cosx+C
D
sin2x/2+C
16
Fundamental Theorem of Calculus (Part 1): d/dx[∫_{a}^{x}f(t)dt] =
A
f(a)
B
∫f(x)dx
C
f(x)
D
F(a)−F(b)
17
Fundamental Theorem (Part 2): ∫_{a}^{b}f(x)dx =
A
F(a)−F(b)
B
F(b)−F(a)
C
F(b)+F(a)
D
f(b)−f(a)
18
Integration by parts: ∫u dv =
A
uv+∫v du
B
uv−∫v du
C
u dv−v du
D
u+v
19
∫x e^x dx (u=x, dv=e^x dx):
A
xe^x+e^x+C
B
xe^x−e^x+C
C
e^x+C
D
xe^x+C
20
∫lnx dx (u=lnx, dv=dx):
A
lnx/x+C
B
x lnx−x+C
C
x lnx+C
D
lnx+C
21
Area between curve y=f(x) and x-axis from a to b:
A
∫_a^b f(x) dx
B
∫_a^b |f(x)| dx
C
f(b)−f(a)
D
F(b)×F(a)
22
Area between y=x² and y=x: intersection at x=0 and x=1. Area=
A
∫₀¹(x−x²)dx=1/2−1/3=1/6
B
∫₀¹(x²−x)dx
C
1/2
D
1/3
23
∫tan x dx =
A
−ln|cosx|+C
B
ln|cosx|+C
C
ln|sinx|+C
D
−ln|sinx|+C
24
∫cot x dx =
A
−ln|sinx|+C
B
ln|sinx|+C
C
ln|cosx|+C
D
−ln|cosx|+C
25
∫sec x dx =
A
ln|secx+tanx|+C
B
secx tanx+C
C
tan²x+C
D
ln|secx|+C
26
∫dx/(1+x²) =
A
ln(1+x²)+C
B
tan⁻¹x+C
C
sin⁻¹x+C
D
sec⁻¹x+C
27
∫dx/√(1−x²) =
A
tan⁻¹x+C
B
ln|x|+C
C
sin⁻¹x+C
D
−cos⁻¹x+C
28
∫₁² 1/x dx =
A
ln2−ln1=ln2
B
1
C
0
D
2
29
∫(2x+3)^4 dx (u=2x+3):
A
(2x+3)^5/10+C
B
(2x+3)^5/5+C
C
(2x+3)^5+C
D
5(2x+3)^4+C
30
∫₀^{π/2} cosx dx =
A
0
B
π/2
C
1
D
2
31
∫₀^{π/2} sin x dx =
A
0
B
1
C
π/2
D
2
32
The constant of integration C represents:
A
A specific value
B
An arbitrary constant (family of functions)
C
Zero
D
A function
33
∫(x²+3x−1)dx =
A
x³/3+3x²/2−x+C
B
2x+3+C
C
x³+3x²−x+C
D
x³/3+3x−1+C
34
∫_{-1}^{1} x³ dx =
A
0
B
2
C
1
D
1/4
35
∫_{-1}^{1} x² dx =
A
0
B
2/3
C
1
D
2
36
Volume of solid of revolution about x-axis: V = π∫_a^b [f(x)]² dx. This is called:
A
Simpson's rule
B
Disk method
C
Shell method
D
Trapezoidal rule
37
∫e^{2x} dx =
A
2e^{2x}+C
B
e^{2x}/2+C
C
e^{2x}+C
D
xe^{2x}+C
38
∫sin(3x) dx =
A
cos(3x)+C
B
−cos(3x)/3+C
C
3cos(3x)+C
D
−3cos(3x)+C
39
∫cos(2x) dx =
A
sin(2x)+C
B
sin(2x)/2+C
C
2sin(2x)+C
D
−sin(2x)/2+C
40
∫x cosx dx (by parts: u=x, dv=cosxdx):
A
x sinx+cosx+C
B
x sinx−cosx+C
C
sinx+x cosx+C
D
x cosx+sinx+C
41
∫x² lnx dx (u=lnx, dv=x²dx):
A
x³lnx/3−x³/9+C
B
x³lnx/3+x³/9+C
C
x³lnx/3+C
D
lnx/x+C
42
The average value of f(x) on [a,b]:
A
f(b)−f(a)
B
(1/(b−a))∫_a^b f(x)dx
C
∫_a^b f(x)dx
D
(f(a)+f(b))/2
43
∫₀¹ x² dx =
A
1/2
B
1/3
C
2/3
D
1
44
∫₀^{π} x sinx dx (by parts):
A
π
B
0
C
2
D
π/2
45
∫dx/x² = ∫x^{-2} dx =
A
−1/x+C
B
1/x+C
C
2/x+C
D
x^{-1}/ln|x|+C
46
∫(sinx/cosx) dx = ∫tanx dx =
A
−ln|cosx|+C
B
ln|cosx|+C
C
ln|sinx|+C
D
secx+C
47
Trapezoidal rule for area: A≈(h/2)[f(x₀)+2f(x₁)+2f(x₂)+...+f(xₙ)] is a:
A
Exact method
B
Numerical approximation
C
Substitution method
D
Integration by parts
48
Simpson's rule requires the number of subintervals to be:
A
Any positive integer
B
Even
C
Odd
D
Prime
49
∫x/(x²+1) dx =
A
ln(x²+1)/2+C
B
arctan(x)+C
C
1/(x²+1)+C
D
2x ln(x²+1)+C
50
∫sin²x cos x dx (u=sinx):
A
sin³x/3+C
B
−cos³x/3+C
C
sinx cosx+C
D
sin²x/2+C
51
The improper integral ∫₁^∞ 1/x dx =
A
1
B
Converges to ln∞=∞ (diverges)
C
0
D
e
52
The improper integral ∫₁^∞ 1/x² dx =
A
Converges to 1
B
Diverges
C
∞
D
0
53
∫₀² 3x² dx =
A
8
B
12
C
6
D
4
54
∫₀^{π/4} sec²x dx =
A
1
B
0
C
tan(π/4)=1
D
π/4
55
If F'(x)=f(x), then ∫f(x)dx =
A
F'(x)+C
B
f(x)+C
C
F(x)+C
D
f'(x)+C
56
∫(3/x + 4x + 5) dx =
A
3ln|x|+2x²+5x+C
B
3x²+2x²+5+C
C
ln|x|/3+4+5x+C
D
3lnx+4x²+5x+C
57
∫₀^1 (1+x)^2 dx =
A
7/3
B
8/3
C
3
D
1/3
58
∫1/(a²+x²) dx =
A
(1/a)arctan(x/a)+C
B
arctan(x)+C
C
arctan(ax)+C
D
(1/a²)arctan(x)+C
59
∫1/√(a²−x²) dx =
A
arcsin(x/a)+C
B
arccos(x/a)+C
C
arctan(x/a)+C
D
1/a arcsin(x)+C
60
Area under y=sinx from 0 to π:
A
0
B
1
C
2
D
π
61
Area bounded by y=x² and y=4 (from −2 to 2):
A
16/3
B
32/3
C
16
D
8
62
∫sin(x+π/4)dx =
A
−cos(x+π/4)+C
B
cos(x+π/4)+C
C
sin(x+π/4)/cos+C
D
−sin(x+π/4)+C
63
Property: ∫_a^a f(x)dx =
A
2f(a)
B
f(a)
C
0
D
∞
64
Property: ∫_a^b f(x)dx = −∫_b^a f(x)dx:
A
True
B
False
C
Only for positive f
D
Only for odd f
65
∫₋₂² x³ dx using odd function property:
A
0
B
16
C
−16
D
8
66
∫tan²x dx = ∫(sec²x−1)dx =
A
tanx−x+C
B
sec²x+C
C
tanx+C
D
sec³x/3−x+C
67
∫csc²x dx =
A
cotx+C
B
−cotx+C
C
cscx+C
D
secx+C
68
∫_{−π}^{π} sinx dx =
A
0
B
2
C
−2
D
π
69
∫_{0}^{π/2} sin²x dx = π/4 (verify: ∫(1-cos2x)/2 dx from 0 to π/2):
A
[x/2−sin2x/4]₀^{π/2}=π/4
B
π/2
C
1
D
π/4. ✓
70
∫e^{-x} dx =
A
e^{-x}+C
B
−e^{-x}+C
C
xe^{-x}+C
D
−xe^{-x}+C
71
∫₀^∞ e^{-x} dx (improper):
A
Diverges
B
0
C
1
D
∞
72
∫(x+1)e^x dx (by parts or notice (xe^x)'=e^x+xe^x):
A
xe^x+C
B
xe^x+e^x+C=(x+1)e^x+C
C
(x+1)e^x+e^x+C
D
xe^x−e^x+C
73
∫√x dx =
A
x^{3/2}+C
B
2x^{3/2}/3+C
C
x^{1/2}/2+C
D
√x/2+C
74
∫₁^4 √x dx =
A
14/3
B
16/3
C
28/3
D
8
75
∫(x²+x+1)/x dx = ∫(x+1+1/x)dx =
A
x²/2+x+ln|x|+C
B
x+1+lnx+C
C
x²/2+x+C
D
x+ln|x|+C
76
∫₀^1 x/(1+x²) dx =
A
ln2/2
B
ln2
C
1/2
D
0
77
∫sin x cos x dx = ∫sin(2x)/2 dx =
A
−cos(2x)/4+C
B
cos(2x)/4+C
C
sin²x+C
D
sin(2x)/4+C
78
∫₀^{π/2} cos(2x)dx =
A
1
B
0
C
π/2
D
π
79
Integration of f(ax+b): ∫f(ax+b)dx = F(ax+b)/a+C. For ∫sin(2x+1)dx:
A
−cos(2x+1)/2+C
B
−cos(2x+1)+C
C
cos(2x+1)/2+C
D
2cos(2x+1)+C
0
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