Q. Calculate the area under the curve y = 2x + 1 from x = 1 to x = 4.
Solution
The area under the curve is given by ∫(from 1 to 4) (2x + 1) dx = [x^2 + x] from 1 to 4 = (16 + 4) - (1 + 1) = 20.
Correct Answer:
A
— 15
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Q. Calculate the area under the curve y = x^3 from x = 0 to x = 2.
Solution
The area under the curve is given by ∫(from 0 to 2) x^3 dx = [x^4/4] from 0 to 2 = (16/4) - (0) = 4.
Correct Answer:
B
— 8
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Q. Calculate the integral ∫(2 to 3) (x^3) dx. (2023)
Solution
∫(2 to 3) (x^3) dx = [x^4/4] from 2 to 3 = (81/4 - 16/4) = 65/4 = 16.25.
Correct Answer:
C
— 8
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Q. Calculate the integral ∫(2 to 5) (4x - 1) dx. (2023)
Solution
∫(2 to 5) (4x - 1) dx = [2x^2 - x] from 2 to 5 = (50 - 5) - (8 - 2) = 40.
Correct Answer:
A
— 20
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Q. Determine the solution of the differential equation dy/dx = y^2 - 1.
-
A.
y = tan(x + C)
-
B.
y = 1/(C - x)
-
C.
y = 1/(C + x)
-
D.
y = e^(x + C)
Solution
This is separable. Separating and integrating gives y = 1/(C - x).
Correct Answer:
B
— y = 1/(C - x)
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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 ∫(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 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) (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 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 ∫(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. Find the area between the curves y = x and y = x^2 from x = 0 to x = 1.
-
A.
0.5
-
B.
1
-
C.
0.25
-
D.
0.75
Solution
The area between the curves is given by ∫(from 0 to 1) (x - x^2) dx = [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.
Correct Answer:
A
— 0.5
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Q. Find the area under the curve y = 3x^2 from x = 1 to x = 2.
Solution
The area under the curve is given by ∫(from 1 to 2) 3x^2 dx = [x^3] from 1 to 2 = (8 - 1) = 7.
Correct Answer:
B
— 6
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Q. Find the general solution of the differential equation dy/dx = 3x^2.
-
A.
y = x^3 + C
-
B.
y = 3x^3 + C
-
C.
y = x^2 + C
-
D.
y = 3x^2 + C
Solution
Integrating both sides gives y = (3/3)x^3 + C = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Find the general solution of the differential equation dy/dx = 4y.
-
A.
y = Ce^(4x)
-
B.
y = 4Ce^x
-
C.
y = Ce^(x/4)
-
D.
y = 4Ce^(x)
Solution
This is a separable differential equation. Integrating gives y = Ce^(4x), where C is the constant.
Correct Answer:
A
— y = Ce^(4x)
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Q. Find the general solution of the equation dy/dx = 3x^2y.
-
A.
y = Ce^(x^3)
-
B.
y = Ce^(3x^3)
-
C.
y = Ce^(x^3/3)
-
D.
y = Ce^(x^2)
Solution
This is a separable equation. Separating and integrating gives y = Ce^(x^3).
Correct Answer:
A
— y = Ce^(x^3)
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Q. Find the integral of (1/x) dx.
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
x + C
-
.
1/x + C
-
.
e^x + C
Solution
The integral of (1/x) is ln|x| + C, where C is the constant of integration.
Correct Answer:
A
— ln
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Q. Find the integral of e^(2x) dx.
-
A.
(1/2)e^(2x) + C
-
B.
2e^(2x) + C
-
C.
e^(2x) + C
-
D.
(1/2)e^(x) + C
Solution
The integral of e^(2x) is (1/2)e^(2x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. Find the integral of x^2 with respect to x.
-
A.
(1/3)x^3 + C
-
B.
(1/2)x^3 + C
-
C.
(1/4)x^4 + C
-
D.
x^3 + C
Solution
The integral of x^2 is (1/3)x^3 + C, where C is the constant of integration.
Correct Answer:
A
— (1/3)x^3 + C
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Q. Find the particular solution of dy/dx = 2y with the initial condition y(0) = 1.
-
A.
y = e^(2x)
-
B.
y = e^(2x) + 1
-
C.
y = 1 + e^(2x)
-
D.
y = e^(2x) - 1
Solution
The general solution is y = Ce^(2x). Using the initial condition y(0) = 1 gives C = 1, so y = e^(2x).
Correct Answer:
A
— y = e^(2x)
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Q. Find the value of the definite integral ∫(0 to 1) (x^2 + 2x) dx. (2020)
Solution
∫(0 to 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. Find the value of the definite integral ∫(0 to 2) (x^2 + 1) dx. (2020)
Solution
∫(0 to 2) (x^2 + 1) dx = [x^3/3 + x] from 0 to 2 = (8/3 + 2) - (0) = 4.
Correct Answer:
B
— 6
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Q. Find the value of the definite integral ∫(0 to π) sin(x) dx. (2019)
Solution
∫(0 to π) sin(x) dx = [-cos(x)] from 0 to π = [1 - (-1)] = 2.
Correct Answer:
C
— 2
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