Pure Mathematics - Integration Practice Questions
A-Levels · A-Level Mathematics · 146 free MCQs with instant results and detailed explanations.
146
Total
43
Easy
73
Medium
30
Hard
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Sample Questions from Pure Mathematics - Integration
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Q1
Easy
What is the integral of the function f(x) = 3x^2 with respect to x?
A.
x^3 + C
B.
3x^3 + C
C.
x^2 + C
D.
6x + C
Show Answer & Explanation
Correct Answer: B
The integral of 3x^2 is found by increasing the power of x by one and dividing by the new power. This gives (3x^(2+1))/(2+1) = 3x^3/3 = x^3, plus the constant of integration C, so the final answer is 3x^3 + C.
Q2
Easy
What is the integral of f(x) = e^x with respect to x?
A.
e^x + C
B.
x e^x + C
C.
e^(x+1) + C
D.
xe^x
Show Answer & Explanation
Correct Answer: A
The integral of e^x with respect to x is a fundamental result in calculus. The integral retains the same form as the original function, plus the constant of integration C. Hence, the answer is e^x + C.
Q3
Easy
What is the integral of the function f(x) = 3x^2 with respect to x?
A.
x^3 + C
B.
x^2 + C
C.
6x + C
D.
3x^3 + C
Show Answer & Explanation
Correct Answer: A
The integral of 3x^2 is found by increasing the exponent by 1 and dividing by the new exponent. Thus, the result is (3/3)x^3 + C = x^3 + C.
Q4
Medium
What is the integral of the function f(x) = 2x^3 - 4x + 1?
A.
0.5x^4 - 2x^2 + x + C
B.
0.5x^4 - 2x^2 + 1 + C
C.
0.5x^4 + 2x^2 + x + C
D.
0.5x^4 - 2x^2 - x + C
Show Answer & Explanation
Correct Answer: A
The integral of 2x^3 is 0.5x^4, the integral of -4x is -2x^2, and the integral of 1 is x, so the correct result is 0.5x^4 - 2x^2 + x + C.
Q5
Medium
Calculate the area under the curve y = x^2 from x = 1 to x = 3.
A.
8
B.
9
C.
10
D.
12
Show Answer & Explanation
Correct Answer: A
The area under the curve is found by calculating the definite integral ∫(from 1 to 3) x^2 dx, which evaluates to (1/3)(3^3 - 1^3) = 8.
Q6
Medium
Evaluate the definite integral ∫(1 to 3) (4x - 1) dx.
A.
14
B.
16
C.
10
D.
12
Show Answer & Explanation
Correct Answer: A
First, find the indefinite integral of 4x - 1, which is 2x^2 - x. Evaluating at the limits 3 and 1 gives: [2(3^2) - 3] - [2(1^2) - 1] = (18 - 3) - (2 - 1) = 15 - 1 = 14.
Q7
Medium
Consider the function f(x) = 2sin(x) + cos(2x). What is the indefinite integral of f(x)?
A.
-2cos(x) + (1/2)sin(2x) + C
B.
2cos(x) + (1/2)sin(2x) + C
C.
-2sin(x) + (1/2)sin(2x) + C
D.
2sin(x) + (1/2)cos(2x) + C
Show Answer & Explanation
Correct Answer: A
The integral of 2sin(x) is -2cos(x), and the integral of cos(2x) is (1/2)sin(2x). Thus, combining these gives the indefinite integral as -2cos(x) + (1/2)sin(2x) + C.
Q8
Hard
Evaluate the integral ∫ (3x^2 - 4x + 5) dx from x = 1 to x = 3.
A.
22
B.
24
C.
20
D.
18
Show Answer & Explanation
Correct Answer: B
The integral evaluates to 24. By finding the antiderivative (x^3 - 2x^2 + 5x) and then calculating its value from 1 to 3, we get (27 - 18 + 15) - (1 - 2 + 5) = 24.
Q9
Hard
Find the area under the curve y = x^2 - 4 from x = -2 to x = 2.
A.
16/3
B.
8/3
C.
0
D.
4
Show Answer & Explanation
Correct Answer: C
The area under the curve from -2 to 2 is 0 because the curve y = x^2 - 4 intersects the x-axis at those points. The positive and negative areas cancel each other out.
Q10
Hard
Find the value of k such that the integral ∫(2x + k) dx from x = 0 to x = 2 is equal to 10.
A.
4
B.
6
C.
8
D.
10
Show Answer & Explanation
Correct Answer: A
To find k, we first calculate the integral: ∫(2x + k) dx = (x^2 + kx) evaluated from 0 to 2. Setting this equal to 10 leads us to k = 4.
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Pure Mathematics - Integration — A-Levels A-Level Mathematics Practice Questions Online
This page contains 146 practice MCQs for the chapter Pure Mathematics - Integration in A-Levels A-Level Mathematics. The questions are organized by difficulty — 43 easy, 73 medium, 30 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.