Q. Evaluate the integral ∫ (3x^2 - 4) dx.
-
A.
x^3 - 4x + C
-
B.
x^3 - 2x + C
-
C.
3x^3 - 4x + C
-
D.
x^3 - 4x
Solution
The integral evaluates to x^3 - 4x + C, where C is the constant of integration.
Correct Answer:
A
— x^3 - 4x + C
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Q. Evaluate the integral ∫ (4x^3 - 2x) dx.
-
A.
x^4 - x^2 + C
-
B.
x^4 - x^2
-
C.
x^4 - x^2 + 2C
-
D.
4x^4 - x^2 + C
Solution
The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.
Correct Answer:
A
— x^4 - x^2 + C
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Q. Evaluate the integral ∫ (5x^4) dx.
-
A.
x^5 + C
-
B.
x^5 + 5C
-
C.
x^5 + 1
-
D.
5x^5 + C
Solution
The integral is (5/5)x^5 + C = x^5 + C.
Correct Answer:
A
— x^5 + C
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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
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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
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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
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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
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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
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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
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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
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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
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Q. Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
∫(0 to 1) (1 - x^2) dx = [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer:
C
— 2/3
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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
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Q. Evaluate the integral ∫(0 to π) sin(x) dx. (2021)
Solution
∫(0 to π) sin(x) dx = [-cos(x)] from 0 to π = -(-1 - 1) = 2.
Correct Answer:
C
— 2
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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
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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
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Q. Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
Solution
∫(1 to 2) (3x^2 - 4) dx = [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.
Correct Answer:
A
— 1
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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
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Q. Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)
Solution
∫(1 to 3) (3x^2 - 2) dx = [x^3 - 2x] from 1 to 3 = (27 - 6) - (1 - 2) = 20.
Correct Answer:
B
— 12
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Q. Evaluate the integral ∫(1 to 4) (2x + 1) dx. (2021)
Solution
∫(1 to 4) (2x + 1) dx = [x^2 + x] from 1 to 4 = (16 + 4) - (1 + 1) = 18.
Correct Answer:
B
— 12
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Q. Evaluate the integral ∫(2 to 3) (x^3 - 3x^2 + 2) dx. (2023)
Solution
∫(2 to 3) (x^3 - 3x^2 + 2) dx = [x^4/4 - x^3 + 2x] from 2 to 3 = (81/4 - 27 + 6) - (16/4 - 8 + 4) = 1.
Correct Answer:
B
— 2
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Q. Evaluate the integral ∫(2x + 3) dx from 1 to 2.
Solution
The integral evaluates to [x^2 + 3x] from 1 to 2, which gives (4 + 6) - (1 + 3) = 8 - 4 = 4.
Correct Answer:
B
— 7
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Q. Evaluate the integral ∫(2x + 3) dx. (2021)
-
A.
x^2 + 3x + C
-
B.
x^2 + 3x
-
C.
2x^2 + 3x + C
-
D.
2x^2 + 3x
Solution
The integral of (2x + 3) is (2x^2/2) + 3x + C = x^2 + 3x + C.
Correct Answer:
A
— x^2 + 3x + C
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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
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Q. Evaluate the integral ∫(3x^2 + 2)dx. (2022)
-
A.
x^3 + 2x + C
-
B.
x^3 + 2x^2 + C
-
C.
x^3 + 2x^3 + C
-
D.
3x^3 + 2x + C
Solution
Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Correct Answer:
A
— x^3 + 2x + C
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Q. Evaluate the integral ∫(sin x)dx. (2022)
-
A.
-cos x + C
-
B.
cos x + C
-
C.
sin x + C
-
D.
-sin x + C
Solution
The integral of sin x is -cos x + C.
Correct Answer:
A
— -cos x + C
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Q. Evaluate the integral ∫(x^2 - 2x + 1) dx. (2022)
-
A.
(1/3)x^3 - x^2 + x + C
-
B.
(1/3)x^3 - x^2 + C
-
C.
(1/3)x^3 - 2x + C
-
D.
(1/3)x^3 - x^2 + x
Solution
The integral of (x^2 - 2x + 1) is (1/3)x^3 - x^2 + x + C.
Correct Answer:
A
— (1/3)x^3 - x^2 + x + C
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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
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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
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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
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