Multiple Choice
Question 1
What is the integral of cos(x) dx?
A
sin(x) + C
B
-sin(x) + C
C
cos(x) + C
D
-cos(x) + C
Solution: The antiderivative of cos(x) is sin(x) + C.
Question 2
What is the integral of a constant k with respect to x?
A
kx + C
B
k + C
C
k^2 x + C
D
0
Solution: The antiderivative of a constant k is kx + C.
Question 3
The integral from 0 to 5 of 0 dx equals:
A
0
B
5
C
1
D
undefined
Solution: The integral of the zero function is 0 over any interval.
Question 4
The Fundamental Theorem of Calculus connects:
A
differentiation and integration as inverse processes
B
limits and continuity
C
sequences and series
D
vectors and matrices
Solution: The FTC establishes that differentiation and integration are inverse operations.
Question 5
If the integral from 1 to 4 of f(x) dx = 10, what is the integral from 4 to 1 of f(x) dx?
A
-10
B
10
C
0
D
5
Solution: Reversing the limits of integration negates the value: -10.
Question 6
If integral from 0 to 2 of f = 3 and integral from 2 to 5 of f = 7, then integral from 0 to 5 of f equals:
A
10
B
4
C
21
D
-4
Solution: By additivity of integrals over adjacent intervals: 3 + 7 = 10.
Question 7
If integral from 0 to 3 of f(x) dx = 8, what is integral from 0 to 3 of 5 f(x) dx?
A
40
B
8
C
13
D
5
Solution: The constant multiple property: integral of 5f = 5 * integral of f = 5 * 8 = 40.
Question 8
Evaluate the integral from 1 to 3 of x^2 dx.
A
26/3
B
9
C
8
D
13
Solution: Antiderivative x^3/3. Evaluate: 27/3 - 1/3 = 26/3.
Question 9
Evaluate the integral from 0 to pi of sin(x) dx.
A
2
B
0
C
1
D
-2
Solution: Antiderivative -cos(x). Evaluate: -cos(pi) - (-cos(0)) = -(-1) - (-1) = 1 + 1 = 2.
Question 10
Evaluate the integral from 1 to e of (1/x) dx.
A
1
B
e
C
e - 1
D
0
Solution: Antiderivative ln|x|. Evaluate: ln(e) - ln(1) = 1 - 0 = 1.
Question 11
Evaluate the integral from 0 to 2 of (3x^2 - 2x) dx.
A
4
B
8
C
6
D
2
Solution: Antiderivative x^3 - x^2. Evaluate: (8 - 4) - 0 = 4.
Question 12
What is the integral of (1/x^2) dx?
A
-1/x + C
B
1/x + C
C
ln|x^2| + C
D
-2/x^3 + C
Solution: Write as x^(-2). Integral = x^(-1)/(-1) + C = -1/x + C.
Question 13
What is the integral of sec^2(x) dx?
A
tan(x) + C
B
sec(x) + C
C
-tan(x) + C
D
cot(x) + C
Solution: Since d/dx tan(x) = sec^2(x), the antiderivative of sec^2(x) is tan(x) + C.
Question 14
What is the integral of e^(2x) dx?
A
(1/2)e^(2x) + C
B
e^(2x) + C
C
2e^(2x) + C
D
e^(2x)/x + C
Solution: The antiderivative of e^(kx) is (1/k)e^(kx) + C. For k = 2: (1/2)e^(2x) + C.
Question 15
Using u-substitution, what is the integral of 2x(x^2 + 1)^3 dx?
A
(x^2 + 1)^4/4 + C
B
(x^2 + 1)^4 + C
C
2(x^2 + 1)^4 + C
D
(x^2 + 1)^3 + C
Solution: Let u = x^2 + 1, du = 2x dx. Integral becomes integral of u^3 du = u^4/4 + C = (x^2 + 1)^4/4 + C.
Question 16
For the integral of cos(3x) dx, the appropriate substitution gives:
A
(1/3)sin(3x) + C
B
sin(3x) + C
C
3 sin(3x) + C
D
-(1/3)sin(3x) + C
Solution: Let u = 3x, du = 3 dx. Integral = (1/3) integral cos(u) du = (1/3)sin(3x) + C.
Question 17
What is the integral of sin(x) cos(x) dx (using u = sin x)?
A
(1/2)sin^2(x) + C
B
sin^2(x) + C
C
-(1/2)cos^2(x) + C only
D
cos(x) + C
Solution: Let u = sin x, du = cos x dx. Integral = integral u du = u^2/2 + C = (1/2)sin^2(x) + C.
Question 18
To integrate (x^2 + x)/x, first simplify to:
A
x + 1
B
x^2 + 1
C
x
D
1/x
Solution: Divide each term by x: (x^2 + x)/x = x + 1, which integrates easily to x^2/2 + x + C.
Question 19
To integrate (x^3 + 2x)/x, rewrite the integrand as:
A
x^2 + 2
B
x^2 + 2x
C
x^3 + 2
D
x + 2
Solution: Divide term by term: (x^3 + 2x)/x = x^2 + 2, which integrates to x^3/3 + 2x + C.
Question 20
What is the integral of 1/(x^2 + 1) dx?
A
arctan(x) + C
B
ln(x^2 + 1) + C
C
(1/2)ln(x^2 + 1) + C
D
-1/(x^2 + 1) + C
Solution: This is a standard form: the antiderivative of 1/(x^2 + 1) is arctan(x) + C.