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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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
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Q. Evaluate ∫ from 0 to 1 of (1 - x^2) dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: C — 2/3
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Q. Evaluate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2) dx.
Solution
The integral evaluates to [x^4 - x^3 + 2x] from 0 to 1 = (1 - 1 + 2) - (0) = 2.
Correct Answer: C — 3
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Q. Evaluate ∫ from 0 to 1 of (x^2 + 3x + 2) dx.
Solution
The integral evaluates to [x^3/3 + (3/2)x^2 + 2x] from 0 to 1 = (1/3 + 3/2 + 2) = (1/3 + 3/2 + 6/3) = 27/6 = 4.5.
Correct Answer: C — 3
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Q. Evaluate ∫ from 0 to 1 of (x^3 + 3x^2 + 3x + 1) dx.
Solution
The integral evaluates to [x^4/4 + x^3 + (3/2)x^2 + x] from 0 to 1 = (1/4 + 1 + 3/2 + 1) = 4.
Correct Answer: D — 4
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Q. Evaluate ∫ from 0 to 1 of (x^4 + 2x^3) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5 + x^4/2] from 0 to 1 = (1/5 + 1/2) = 7/10.
Correct Answer: C — 1/3
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Q. Evaluate ∫ from 0 to 1 of (x^4) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Evaluate ∫ 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 ∫ from 0 to 2 of (x^2 + 2x + 1) dx.
Solution
The integral evaluates to [x^3/3 + x^2 + x] from 0 to 2 = (8/3 + 4 + 2) = 6.
Correct Answer: C — 6
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Q. Evaluate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.
Solution
The integral evaluates to [x^4/4 - x^3 + 4x] from 0 to 2 = (4 - 8 + 8) - 0 = 4.
Correct Answer: C — 6
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Q. Evaluate ∫ from 1 to 2 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.
Solution
The integral evaluates to [x^5/5 - x^4 + 2x^3 - 2x^2 + x] from 1 to 2 = 0.
Correct Answer: A — 0
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Q. Evaluate ∫ 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 — 10
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Q. Evaluate ∫ from 1 to 3 of (x^2 - 4) dx.
Solution
The integral evaluates to [x^3/3 - 4x] from 1 to 3 = (27/3 - 12) - (1/3 - 4) = 2.
Correct Answer: C — 2
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Q. Evaluate ∫_0^1 (1 - x^2) dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
∫_0^1 (1 - x^2) dx = [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: B — 1/2
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Q. Evaluate ∫_0^1 (e^x) dx.
Solution
∫_0^1 e^x dx = [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
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Q. Evaluate ∫_0^1 (x^3 + 2x^2) dx.
-
A.
1/4
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
∫_0^1 (x^3 + 2x^2) dx = [x^4/4 + 2x^3/3] from 0 to 1 = (1/4 + 2/3) = 11/12.
Correct Answer: C — 1/2
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Q. Evaluate ∫_0^1 (x^3 - 3x^2 + 3x - 1) dx.
Solution
The integral evaluates to [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 ∫_0^1 (x^4 - 2x^2 + 1) dx.
Solution
∫_0^1 (x^4 - 2x^2 + 1) dx = [x^5/5 - (2/3)x^3 + x] from 0 to 1 = (1/5 - 2/3 + 1) = 1/15.
Correct Answer: B — 1
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Q. Evaluate ∫_0^1 (x^4) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Evaluate ∫_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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Q. Evaluate ∫_0^π/2 sin^2(x) dx.
-
A.
π/4
-
B.
π/2
-
C.
π/3
-
D.
π/6
Solution
Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.
Correct Answer: A — π/4
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Q. Evaluate ∫_1^2 (3x^2 - 4) dx.
Solution
The integral evaluates to [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.
Correct Answer: A — 1
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Q. Evaluate ∫_1^2 (3x^2 - 4x + 1) dx.
Solution
∫_1^2 (3x^2 - 4x + 1) dx = [x^3 - 2x^2 + x] from 1 to 2 = (8 - 8 + 2) - (1 - 2 + 1) = 1.
Correct Answer: B — 1
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Q. Evaluate ∫_1^3 (2x + 1) dx.
Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer: B — 10
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Q. Find the value of ∫ from 0 to 1 of (1 - x^2) dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: C — 2/3
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Q. Find the value of ∫ 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. Find the value of ∫ from 0 to 1 of (x^2 * e^x) dx.
Solution
Using integration by parts, the integral evaluates to (e - 1)/2.
Correct Answer: B — e - 1
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Q. Find the value of ∫ from 0 to 1 of (x^2 + 1/x^2) dx.
Solution
The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.
Correct Answer: C — 3
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Q. Find the value of ∫ from 0 to 1 of (x^2 + 3x + 2) dx.
Solution
The integral evaluates to [x^3/3 + (3/2)x^2 + 2x] from 0 to 1 = (1/3 + 3/2 + 2) - 0 = 4/3 + 3/2 = 2.5.
Correct Answer: D — 4
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