Q. Evaluate the integral ∫ (3x^2 + 2x + 1) dx.
-
A.
x^3 + x^2 + x + C
-
B.
x^3 + x^2 + C
-
C.
x^3 + x^2 + x
-
D.
3x^3 + 2x^2 + x + C
Solution
The integral of 3x^2 is x^3, the integral of 2x is x^2, and the integral of 1 is x. Therefore, ∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.
Correct Answer: A — x^3 + x^2 + x + C
Learn More →
Q. Evaluate the integral ∫ (sec^2(x)) dx.
-
A.
tan(x) + C
-
B.
sec(x) + C
-
C.
sin(x) + C
-
D.
cos(x) + C
Solution
The integral of sec^2(x) is tan(x) + C.
Correct Answer: A — tan(x) + C
Learn More →
Q. Evaluate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
-
A.
(1/3)x^3 + x^2 + C
-
B.
x^2 + x + C
-
C.
(1/3)x^3 + (1/2)x^2 + C
-
D.
x^2 + 2x + C
Solution
By simplifying the integrand, we can integrate to find that ∫ (x^2 + 2x + 1)/(x + 1) dx = (1/3)x^3 + x^2 + C.
Correct Answer: A — (1/3)x^3 + x^2 + C
Learn More →
Q. Evaluate the integral ∫ cos(3x) dx.
-
A.
(1/3)sin(3x) + C
-
B.
sin(3x) + C
-
C.
(1/3)cos(3x) + C
-
D.
-(1/3)sin(3x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 3, so ∫ cos(3x) dx = (1/3)sin(3x) + C.
Correct Answer: A — (1/3)sin(3x) + C
Learn More →
Q. Evaluate the integral ∫ cos(5x) dx.
-
A.
1/5 sin(5x) + C
-
B.
-1/5 sin(5x) + C
-
C.
5 sin(5x) + C
-
D.
sin(5x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 5, so ∫ cos(5x) dx = (1/5)sin(5x) + C.
Correct Answer: A — 1/5 sin(5x) + C
Learn More →
Q. Evaluate the integral ∫ e^(3x) dx.
-
A.
(1/3)e^(3x) + C
-
B.
(1/3)e^(3x)
-
C.
3e^(3x) + C
-
D.
e^(3x) + C
Solution
The integral of e^(kx) is (1/k)e^(kx). Here, k = 3, so ∫ e^(3x) dx = (1/3)e^(3x) + C.
Correct Answer: A — (1/3)e^(3x) + C
Learn More →
Q. Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.
Solution
The integral evaluates to [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.
Correct Answer: B — 2
Learn More →
Q. Evaluate the integral ∫ from 0 to 1 of e^x dx.
Solution
The integral evaluates to [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
Learn More →
Q. Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.
Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer: B — 8
Learn More →
Q. Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.
-
A.
1/4
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
The integral evaluates to [x^4/4 + 2x^3/3] from 0 to 1 = 1/4 + 2/3 = 11/12.
Correct Answer: B — 1/3
Learn More →
Q. Evaluate the integral ∫(1 to 2) (2x + 3)dx.
Solution
∫(2x + 3)dx = [x^2 + 3x] from 1 to 2 = (4 + 6) - (1 + 3) = 10 - 4 = 6.
Correct Answer: B — 8
Learn More →
Q. Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.
Solution
The integral evaluates to [(x^3 - 2x)] from 1 to 2 = (8 - 4) - (1 - 2) = 5.
Correct Answer: A — 3
Learn More →
Q. Evaluate the integral ∫(1 to 2) (x^2 + 2x)dx.
Solution
The integral ∫(x^2 + 2x)dx = [(1/3)x^3 + x^2] from 1 to 2 = 8.
Correct Answer: B — 8
Learn More →
Q. Evaluate the integral ∫(2x^3 - 4x)dx.
-
A.
(1/2)x^4 - 2x^2 + C
-
B.
(1/4)x^4 - 2x^2 + C
-
C.
(1/2)x^4 - 4x^2 + C
-
D.
(1/3)x^4 - 2x^2 + C
Solution
The integral ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.
Correct Answer: A — (1/2)x^4 - 2x^2 + C
Learn More →
Q. Evaluate the integral ∫_0^1 (x^2 + 2x) dx.
Solution
∫_0^1 (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.
Correct Answer: B — 2
Learn More →
Q. Evaluate the integral ∫_0^1 (x^3 - 3x^2 + 3x - 1) dx.
Solution
∫_0^1 (x^3 - 3x^2 + 3x - 1) dx = [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.
Correct Answer: A — 0
Learn More →
Q. Evaluate the integral ∫_0^π/2 cos^2(x) dx.
Solution
∫_0^π/2 cos^2(x) dx = π/4.
Correct Answer: A — π/4
Learn More →
Q. Evaluate the integral ∫_1^2 (3x^2 - 2) dx.
Solution
∫_1^2 (3x^2 - 2) dx = [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.
Correct Answer: A — 1
Learn More →
Q. Evaluate the integral: ∫ (1/(x^2 + 1)) dx
-
A.
tan^(-1)(x) + C
-
B.
sin^(-1)(x) + C
-
C.
ln
-
D.
x
-
.
+ C
-
.
cos^(-1)(x) + C
Solution
The integral of 1/(x^2 + 1) is tan^(-1)(x) + C.
Correct Answer: A — tan^(-1)(x) + C
Learn More →
Q. Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx
-
A.
(1/2)x^4 - x^3 + 4x + C
-
B.
(1/4)x^4 - (1/3)x^3 + 4x + C
-
C.
(1/2)x^4 - (1/3)x^3 + 4x + C
-
D.
(1/4)x^4 - x^3 + 4x + C
Solution
Integrating term by term gives (1/4)x^4 - (1/3)x^3 + 4x + C.
Correct Answer: A — (1/2)x^4 - x^3 + 4x + C
Learn More →
Q. Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).
Solution
Factoring gives (x-1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim x->1 (x^2 + x + 1) = 3.
Correct Answer: C — 2
Learn More →
Q. Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
Solution
Factoring gives (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1, thus limit is 3.
Correct Answer: C — 3
Learn More →
Q. Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).
Solution
Factoring gives (x-2)(x+2)/(x-2). Canceling gives lim x->2 (x + 2) = 4.
Correct Answer: C — 2
Learn More →
Q. Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).
-
A.
0
-
B.
2
-
C.
4
-
D.
undefined
Solution
Factoring gives (x-2)(x+2)/(x-2) = x + 2, thus limit is 4.
Correct Answer: C — 4
Learn More →
Q. Evaluate the limit lim(x→∞) (3x^2 + 2)/(5x^2 - 4).
Solution
Dividing by x^2, lim(x→∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: A — 3/5
Learn More →
Q. Evaluate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Undefined
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1/2.
Correct Answer: B — 1/2
Learn More →
Q. Evaluate the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
e
-
D.
Infinity
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator: lim (x -> 0) (e^x)/(1) = e^0 = 1.
Correct Answer: B — 1
Learn More →
Q. Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Using L'Hôpital's Rule, differentiate the numerator and denominator: lim (x -> 0) (1/(1 + x))/(1) = 1.
Correct Answer: B — 1
Learn More →
Q. Evaluate the limit: lim (x -> 0) (sin(5x)/x)
-
A.
0
-
B.
5
-
C.
1
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (sin(x)/x) = 1, we have lim (x -> 0) (sin(5x)/x) = 5 * lim (x -> 0) (sin(5x)/(5x)) = 5 * 1 = 5.
Correct Answer: B — 5
Learn More →
Q. Evaluate the limit: lim (x -> 0) (tan(3x)/x)
-
A.
0
-
B.
3
-
C.
1
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3 * 1 = 3.
Correct Answer: B — 3
Learn More →
Showing 1771 to 1800 of 10700 (357 Pages)
