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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Q. Find the value of ∫ from 0 to 1 of (x^2 - 2x + 1) dx.
Solution
The integral evaluates to [x^3/3 - x^2 + x] from 0 to 1 = (1/3 - 1 + 1) = 1/3.
Correct Answer: A — 0
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 2) dx.
Solution
The integral evaluates to [x^4/4 - x^3 + 2x] from 0 to 1 = (1/4 - 1 + 2) - (0) = 1/4.
Correct Answer: B — 1
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Q. Find the value of ∫ 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 = 0.
Correct Answer: A — 0
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 3x) dx.
Solution
The integral evaluates to [x^4/4 - x^3 + (3/2)x^2] from 0 to 1 = (1/4 - 1 + 3/2) = 1/4.
Correct Answer: B — 1
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 4x + 4) dx.
Solution
The integral evaluates to [x^4/4 - 2x^2 + 4x] from 0 to 1 = (1/4 - 2 + 4) = (1/4 + 2) = 9/4.
Correct Answer: B — 1
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Q. Find the value of ∫ from 0 to 1 of (x^4 + 2x^2) dx.
-
A.
1/5
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
The integral evaluates to [x^5/5 + (2/3)x^3] from 0 to 1 = (1/5 + 2/3) = 11/15.
Correct Answer: B — 1/3
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Q. Find the value of ∫ from 0 to 1 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 0 to 1 = (1/5 - 1 + 2 - 2 + 1) = 0.
Correct Answer: B — 1
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Q. Find the value of ∫ 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. Find the value of ∫ 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) = 2/3.
Correct Answer: C — 2
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Q. Find the value of ∫ from 1 to 2 of (3x^2 - 2) dx.
Solution
The integral evaluates to [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 4 + 1 = 5.
Correct Answer: A — 1
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Q. Find the value of ∫ from 1 to 2 of (3x^2 - 2x + 1) dx.
Solution
The integral evaluates to [x^3 - x^2 + x] from 1 to 2 = (8 - 4 + 2) - (1 - 1 + 1) = 5.
Correct Answer: C — 5
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Q. Find the value of ∫_0^1 (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: B — 1/2
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Q. Find the value of ∫_0^1 (4x^3) dx.
Solution
∫_0^1 (4x^3) dx = [x^4] from 0 to 1 = 1.
Correct Answer: A — 1
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Q. Find the value of ∫_0^1 (x^2 + 1) dx.
Solution
∫_0^1 (x^2 + 1) dx = [x^3/3 + x] from 0 to 1 = (1/3 + 1) = 4/3.
Correct Answer: B — 2
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Q. Find the value of ∫_0^1 (x^4 + 2x^3 + x^2) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5 + (1/2)x^4 + (1/3)x^3] from 0 to 1 = 1/5 + 1/2 + 1/3 = 31/30.
Correct Answer: B — 1/4
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Q. Find the value of ∫_0^1 (x^4 + 2x^3) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
∫_0^1 (x^4 + 2x^3) dx = [x^5/5 + (1/2)x^4] from 0 to 1 = (1/5 + 1/2) = 7/10.
Correct Answer: A — 1/5
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Q. Find the value of ∫_0^1 (x^4) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
∫_0^1 x^4 dx = [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Find the value of ∫_0^π sin(x) cos(x) dx.
Solution
Using the identity sin(2x) = 2sin(x)cos(x), the integral becomes (1/2)∫_0^π sin(2x) dx = 0.
Correct Answer: A — 0
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Q. Find the value of ∫_0^π sin(x) dx.
Solution
∫_0^π sin(x) dx = [-cos(x)] from 0 to π = -(-1 - 1) = 2.
Correct Answer: C — 2
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Q. Find the value of ∫_0^π/2 cos^2(x) dx.
Solution
The integral evaluates to [x/2 + sin(2x)/4] from 0 to π/2 = π/4.
Correct Answer: A — π/4
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Q. Find the values of x that satisfy 3cos^2(x) - 1 = 0.
-
A.
π/3, 2π/3
-
B.
0, π
-
C.
π/2, 3π/2
-
D.
0, 2π
Solution
Solving gives cos^2(x) = 1/3, so x = π/3, 2π/3, and their equivalents.
Correct Answer: A — π/3, 2π/3
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Q. Find the values of x that satisfy 3sin(x) - 1 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
7π/6
-
D.
11π/6
Solution
The solution is x = π/6 + 2nπ.
Correct Answer: A — π/6
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Q. Find the values of x that satisfy sin^2(x) - sin(x) - 2 = 0.
-
A.
-1, 2
-
B.
1, -2
-
C.
2, -1
-
D.
0, 1
Solution
Factoring gives (sin(x) - 2)(sin(x) + 1) = 0, so sin(x) = 2 (not possible) or sin(x) = -1, giving x = 3π/2.
Correct Answer: A — -1, 2
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Q. Find the values of x that satisfy sin^2(x) - sin(x) = 0.
-
A.
0, π
-
B.
0, π/2
-
C.
0, 2π
-
D.
0, 3π/2
Solution
Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0 and x = π.
Correct Answer: A — 0, π
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Q. Find the values of x that satisfy the equation 3sin(x) - 1 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
7π/6
-
D.
11π/6
Solution
Rearranging gives sin(x) = 1/3, which has solutions in the specified interval.
Correct Answer: A — π/6
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Q. Find the values of x that satisfy the equation 3sin(x) - 2 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
π/2
-
D.
7π/6
Solution
The solution is x = π/2.
Correct Answer: C — π/2
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Q. Find the values of x that satisfy the equation sin^2(x) - sin(x) = 0.
-
A.
0, π
-
B.
0, π/2
-
C.
0, 2π
-
D.
0, 3π/2
Solution
Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0 and x = π.
Correct Answer: A — 0, π
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