Multiple Choice
Question 1
Which technique is most appropriate for the integral of x cos(x^2) dx?
A
u-substitution with u = x^2
B
long division
C
direct power rule
D
no technique needed
Solution: Because the derivative of x^2 (namely 2x) appears (up to a constant), u = x^2 makes this a straightforward substitution.
Question 2
The integral of a sum of functions equals:
A
the sum of the integrals
B
the product of the integrals
C
always zero
D
the difference of the integrals
Solution: Integration is linear: the integral of (f + g) equals the integral of f plus the integral of g.
Question 3
Let g(x) = integral from 0 to x of f(t) dt. If f(x) > 0 on an interval, then g is:
A
increasing there
B
decreasing there
C
constant
D
negative
Solution: Since g'(x) = f(x) > 0, the accumulation function g is increasing wherever f is positive.
Question 4
A table gives f(0) = 2, f(2) = 5, f(4) = 6. Estimate the integral from 0 to 4 of f dx with a right Riemann sum (2 subintervals).
A
22
B
14
C
18
D
26
Solution: Width 2, right endpoints f(2) and f(4): 2(5) + 2(6) = 10 + 12 = 22.
Question 5
Evaluate the integral from -1 to 1 of x^3 dx.
A
0
B
2
C
1/2
D
1
Solution: x^3 is an odd function integrated over a symmetric interval [-1, 1], so the integral is 0. (Check: [x^4/4] from -1 to 1 = 1/4 - 1/4 = 0.)
Question 6
What is the integral of (4x^3 - 6x + 5) dx?
A
x^4 - 3x^2 + 5x + C
B
12x^2 - 6 + C
C
x^4 - 6x^2 + 5x + C
D
4x^4 - 3x^2 + 5x + C
Solution: Integrate term by term: 4x^4/4 - 6x^2/2 + 5x = x^4 - 3x^2 + 5x + C.
Question 7
If F(x) = integral from 2 to x of (t^3) dt, then F'(2) equals:
A
8
B
0
C
2
D
16
Solution: By FTC, F'(x) = x^3, so F'(2) = 2^3 = 8.
Question 8
If integral from 1 to 5 of f = 12 and integral from 1 to 3 of f = 4, then integral from 3 to 5 of f equals:
A
8
B
16
C
4
D
48
Solution: By additivity: integral 1 to 5 = integral 1 to 3 + integral 3 to 5, so 12 = 4 + (integral 3 to 5), giving 8.
Question 9
What is the integral of e^(x^2) * 2x dx?
A
e^(x^2) + C
B
2 e^(x^2) + C
C
e^(2x) + C
D
x^2 e^(x^2) + C
Solution: Let u = x^2, du = 2x dx. Integral = integral e^u du = e^u + C = e^(x^2) + C.
Question 10
Evaluate the integral from 1 to 2 of (1/x^2) dx.
A
1/2
B
1
C
-1/2
D
3/2
Solution: Antiderivative -1/x. Evaluate: -1/2 - (-1/1) = -1/2 + 1 = 1/2.
Question 11
Evaluate the integral from 0 to 3 of (2x + 1) dx.
A
12
B
9
C
10
D
6
Solution: Antiderivative x^2 + x. Evaluate: (9 + 3) - 0 = 12.
Question 12
What is the integral of 3/(x^2 + 1) dx?
A
3 arctan(x) + C
B
arctan(3x) + C
C
3 ln(x^2 + 1) + C
D
arctan(x) + C
Solution: The constant factors out: 3 * integral 1/(x^2 + 1) dx = 3 arctan(x) + C.
Question 13
If the velocity of a particle is v(t) = 3t^2, the displacement from t = 0 to t = 2 is:
A
8
B
12
C
4
D
6
Solution: Displacement = integral from 0 to 2 of 3t^2 dt = [t^3] from 0 to 2 = 8.
Question 14
Increasing the number of subintervals in a Riemann sum generally:
A
improves the accuracy of the approximation
B
makes it less accurate
C
has no effect
D
gives zero
Solution: As the number of subintervals increases (width decreases), the Riemann sum approaches the exact value of the definite integral.
Question 15
What is the integral of (ln x)/x dx?
A
(1/2)(ln x)^2 + C
B
ln(ln x) + C
C
(ln x)^2 + C
D
1/x + C
Solution: Let u = ln x, du = (1/x) dx. Integral = integral u du = u^2/2 + C = (1/2)(ln x)^2 + C.
Question 16
What is the integral of sin(x) dx?
A
-cos(x) + C
B
cos(x) + C
C
-sin(x) + C
D
sin(x) + C
Solution: The antiderivative of sin(x) is -cos(x) + C.
Question 17
Let g(x) = integral from 0 to x of f(t) dt. If f is negative on (a, b), then on that interval g is:
A
decreasing
B
increasing
C
concave up
D
constant
Solution: g'(x) = f(x) < 0 there, so g is decreasing wherever f is negative.
Question 18
The width of each subinterval when [a, b] is divided into n equal parts is:
A
(b - a)/n
B
n/(b - a)
C
b - a
D
(a + b)/n
Solution: Dividing an interval of length (b - a) into n equal subintervals gives width delta x = (b - a)/n.
Question 19
Evaluate the integral from 1 to 4 of (2/sqrt(x)) dx.
A
4
B
6
C
2
D
8
Solution: 2/sqrt(x) = 2x^(-1/2). Antiderivative 4x^(1/2). Evaluate: 4(2) - 4(1) = 8 - 4 = 4.
Question 20
If F is an antiderivative of f and F(1) = 3, F(4) = 11, then integral from 1 to 4 of f(x) dx equals:
A
8
B
14
C
3
D
11
Solution: By FTC Part 2: integral = F(4) - F(1) = 11 - 3 = 8.