Q. Find the value of the definite integral ∫(1 to 3) (x^2 - 2x + 1) dx. (2021)
Solution
∫(1 to 3) (x^2 - 2x + 1) dx = [(x^3/3 - x^2 + x)] from 1 to 3 = (9/3 - 9 + 3) - (1/3 - 1 + 1) = 0.
Correct Answer:
A
— 0
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Q. Find the value of the definite integral ∫(1 to 4) (x^3) dx. (2019)
Solution
∫(1 to 4) (x^3) dx = [x^4/4] from 1 to 4 = (256/4 - 1/4) = 63.
Correct Answer:
C
— 40
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Q. Find the value of the integral ∫(2x + 1)dx from 0 to 2. (2020)
Solution
The integral ∫(2x + 1)dx = x^2 + x. Evaluating from 0 to 2 gives (2^2 + 2) - (0 + 0) = 4.
Correct Answer:
B
— 4
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Q. Solve the differential equation dy/dx = 2y.
-
A.
y = Ce^(2x)
-
B.
y = 2Ce^x
-
C.
y = Ce^(x/2)
-
D.
y = 2x + C
Solution
This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer:
A
— y = Ce^(2x)
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Q. Solve the differential equation dy/dx = 5 - 2y.
-
A.
y = 5/2 + Ce^(-2x)
-
B.
y = 5/2 - Ce^(-2x)
-
C.
y = 2.5 + Ce^(2x)
-
D.
y = 2.5 - Ce^(2x)
Solution
Rearranging gives dy/(5 - 2y) = dx. Integrating both sides leads to y = 5/2 + Ce^(-2x).
Correct Answer:
A
— y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation dy/dx = 6x.
-
A.
y = 3x^2 + C
-
B.
y = 6x^2 + C
-
C.
y = 2x^2 + C
-
D.
y = 3x + C
Solution
Integrating gives y = 3x^2 + C, where C is the constant of integration.
Correct Answer:
A
— y = 3x^2 + C
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Q. Solve the differential equation dy/dx = y^2.
-
A.
y = 1/(C - x)
-
B.
y = Cx
-
C.
y = C + x^2
-
D.
y = C - x
Solution
This is a separable equation. Integrating gives y = 1/(C - x), where C is the constant.
Correct Answer:
A
— y = 1/(C - x)
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Q. What is the area under the curve y = 1/x from x = 1 to x = 4?
-
A.
ln(4)
-
B.
ln(3)
-
C.
ln(2)
-
D.
ln(1)
Solution
The area under the curve is given by ∫(from 1 to 4) (1/x) dx = [ln(x)] from 1 to 4 = ln(4) - ln(1) = ln(4).
Correct Answer:
A
— ln(4)
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Q. What is the area under the curve y = 2x^2 + 3 from x = 0 to x = 2?
Solution
The area under the curve is given by ∫(from 0 to 2) (2x^2 + 3) dx = [(2/3)x^3 + 3x] from 0 to 2 = (16/3 + 6) = 10.
Correct Answer:
B
— 12
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Q. What is 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 = ∫3x^2 dx = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. What is the general solution of the differential equation dy/dx = 4y? (2019)
-
A.
y = Ce^(4x)
-
B.
y = Ce^(x/4)
-
C.
y = 4Ce^x
-
D.
y = 4Ce^(4x)
Solution
The differential equation dy/dx = 4y can be solved using separation of variables, leading to y = Ce^(4x).
Correct Answer:
A
— y = Ce^(4x)
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Q. What is the indefinite integral of e^x? (2020)
-
A.
e^x + C
-
B.
e^x
-
C.
x e^x + C
-
D.
x^2 e^x + C
Solution
The indefinite integral of e^x is e^x + C.
Correct Answer:
A
— e^x + C
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Q. What is the integral of cos(3x) dx?
-
A.
(1/3)sin(3x) + C
-
B.
3sin(3x) + C
-
C.
(1/3)cos(3x) + C
-
D.
sin(3x) + C
Solution
The integral of cos(3x) is (1/3)sin(3x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/3)sin(3x) + C
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Q. What is 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. What is the integral of e^(2x)? (2023)
-
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.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. What is the integral of e^x dx?
-
A.
e^x + C
-
B.
e^x
-
C.
x e^x + C
-
D.
x e^x
Solution
The integral of e^x is e^x + C, where C is the constant of integration.
Correct Answer:
A
— e^x + C
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Q. What is the integral of f(x) = 1/x? (2023)
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
1/x + C
-
.
x + C
-
.
e^x + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer:
A
— ln
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Q. What is the integral of the function f(x) = 3x^2? (2021)
-
A.
x^3 + C
-
B.
x^3 + 3C
-
C.
x^3 + 1
-
D.
3x^3 + C
Solution
The integral of 3x^2 is (3/3)x^3 + C = x^3 + C.
Correct Answer:
A
— x^3 + C
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Q. What is the integral of x^n dx, where n ≠ -1? (2023)
-
A.
(x^(n+1))/(n+1) + C
-
B.
(x^(n-1))/(n-1) + C
-
C.
nx^(n-1) + C
-
D.
x^n + C
Solution
The integral of x^n dx is (x^(n+1))/(n+1) + C.
Correct Answer:
A
— (x^(n+1))/(n+1) + C
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Q. What is the integrating factor for the equation dy/dx + 3y = 6?
-
A.
e^(3x)
-
B.
e^(-3x)
-
C.
3e^(3x)
-
D.
3e^(-3x)
Solution
The integrating factor is e^(∫3dx) = e^(3x).
Correct Answer:
A
— e^(3x)
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Q. What is the solution of the differential equation dy/dx = y/x?
-
A.
y = Cx
-
B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is separable. Integrating gives y = Cx.
Correct Answer:
A
— y = Cx
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Q. What is the solution to the differential equation dy/dx = 3x^2?
-
A.
x^3 + C
-
B.
3x^3 + C
-
C.
x^2 + C
-
D.
3x^2 + C
Solution
Integrating both sides gives y = x^3 + C, where C is the constant of integration.
Correct Answer:
A
— x^3 + C
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Q. What is the solution to the differential equation dy/dx = 6x?
-
A.
y = 3x^2 + C
-
B.
y = 6x^2 + C
-
C.
y = 2x^2 + C
-
D.
y = 3x + C
Solution
Integrating gives y = (6/2)x^2 + C = 3x^2 + C.
Correct Answer:
A
— y = 3x^2 + C
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Q. What is the solution to the differential equation dy/dx = xy?
-
A.
y = Ce^(x^2/2)
-
B.
y = Ce^(-x^2/2)
-
C.
y = Cx^2
-
D.
y = C/x
Solution
This is separable. Separating and integrating gives y = Ce^(x^2/2).
Correct Answer:
A
— y = Ce^(x^2/2)
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Q. What is the solution to the initial value problem dy/dx = 4y, y(1) = 2?
-
A.
y = 2e^(4x)
-
B.
y = 2e^(4x-4)
-
C.
y = e^(4x)
-
D.
y = 4e^(x)
Solution
The general solution is y = Ce^(4x). Using y(1) = 2, we find C = 2e^(-4).
Correct Answer:
B
— y = 2e^(4x-4)
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