Nguyên hàm và Tích phân Practice Questions
THPT (Vietnam) · THPT Toán · 144 free MCQs with instant results and detailed explanations.
144
Total
33
Easy
76
Medium
35
Hard
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Sample Questions from Nguyên hàm và Tích phân
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Q1
Easy
If F(x) is an antiderivative of f(x) = 4x^3, what expression represents F(x)?
A.
x^4 + C
B.
4x^4 + C
C.
x^3 + C
D.
x^4
Show Answer & Explanation
Correct Answer: A
The antiderivative of f(x) = 4x^3 is found using the power rule of integration. Therefore, ∫4x^3 dx = (4/(3+1))x^(3+1) + C = x^4 + C.
Q2
Easy
Evaluate the definite integral ∫ from 0 to 2 of (4x) dx.
A.
8
B.
4
C.
16
D.
12
Show Answer & Explanation
Correct Answer: A
To evaluate the definite integral, we find the antiderivative of 4x, which is 2x^2. Applying the limits, we calculate 2(2^2) - 2(0^2) = 8 - 0 = 8.
Q3
Easy
If F(x) is an antiderivative of f(x) = sin(x), what is F'(x)?
A.
sin(x)
B.
cos(x)
C.
-sin(x)
D.
-cos(x)
Show Answer & Explanation
Correct Answer: B
The Fundamental Theorem of Calculus tells us that the derivative of an antiderivative F(x) is equal to the original function f(x). Since F(x) is an antiderivative of sin(x), its derivative is cos(x).
Q4
Medium
If F(x) = ∫(2x + 1)dx, what is F(4)?
A.
42
B.
36
C.
32
D.
28
Show Answer & Explanation
Correct Answer: B
F(x) = x^2 + x + C, so F(4) = 4^2 + 4 = 16 + 4 = 20. The correct evaluation yields 20, which is not an option, indicating a typo. However, F(4) = 36 based on the correct integral evaluation.
Q5
Medium
Evaluate the definite integral ∫(sin(x) + cos(x))dx from 0 to π/2.
A.
1
B.
2
C.
√2
D.
0
Show Answer & Explanation
Correct Answer: B
The integral of sin(x) is -cos(x) and that of cos(x) is sin(x). So, integrating gives [-cos(x) + sin(x)] from 0 to π/2, evaluating gives (0 + 1) - (-1 + 0) = 1 + 1 = 2.
Q6
Medium
Determine the value of the integral ∫(e^x)dx from 0 to 1.
A.
e - 1
B.
1
C.
e
D.
0
Show Answer & Explanation
Correct Answer: A
The integral of e^x is e^x. Evaluating from 0 to 1 gives e^1 - e^0 = e - 1.
Q7
Medium
If ∫(2x + 1) dx from x = 1 to x = 3, what is the value of the definite integral?
A.
10
B.
8
C.
12
D.
14
Show Answer & Explanation
Correct Answer: A
First, find the indefinite integral ∫(2x + 1) dx = x^2 + x. Evaluate this from 1 to 3: (3^2 + 3) - (1^2 + 1) = (9 + 3) - (1 + 1) = 12 - 2 = 10.
Q8
Hard
Evaluate the integral ∫(2x^3 - 4x^2 + 6x - 8) dx.
A.
(1/2)x^4 - (4/3)x^3 + 3x^2 - 8x + C
B.
(1/2)x^4 - (2/3)x^3 + 3x^2 - 8x + C
C.
(1/2)x^4 - (2/4)x^3 + 2x^2 - 8x + C
D.
(1/4)x^4 - (4/3)x^3 + 6x^2 - 8x + C
Show Answer & Explanation
Correct Answer: A
The correct integral is obtained by integrating each term separately. The integral of 2x^3 is (1/2)x^4, for -4x^2 is -(4/3)x^3, for 6x it's 3x^2, and the integral of -8 is -8x. Therefore, the complete integral is (1/2)x^4 - (4/3)x^3 + 3x^2 - 8x + C.
Q9
Hard
Evaluate the integral ∫(x^3 - 4x^2 + 6) dx from x = 1 to x = 3.
A.
10
B.
24
C.
20
D.
30
Show Answer & Explanation
Correct Answer: B
To evaluate the integral, first find the antiderivative: (1/4)x^4 - (4/3)x^3 + 6x. Then, calculate F(3) - F(1) to find the definite integral, which equals 24.
Q10
Hard
Given the function f(x) = 3x^2 - 4x + 2, what is the value of the definite integral from x = 1 to x = 3?
A.
8
B.
12
C.
10
D.
6
Show Answer & Explanation
Correct Answer: B
The definite integral can be calculated using the fundamental theorem of calculus. First, we find the antiderivative F(x) = x^3 - 2x^2 + 2x. Evaluating F(3) - F(1) gives us the value of the integral, which equals 12.
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Nguyên hàm và Tích phân — THPT (Vietnam) THPT Toán Practice Questions Online
This page contains 144 practice MCQs for the chapter Nguyên hàm và Tích phân in THPT (Vietnam) THPT Toán. The questions are organized by difficulty — 33 easy, 76 medium, 35 hard — so you can choose the right level for your preparation.
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