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Calculus - Integration Practice Questions

Thanaweya Amma (Egypt) · Thanaweya Mathematics · 138 free MCQs with instant results and detailed explanations.

138
Total
32
Easy
75
Medium
31
Hard

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Sample Questions from Calculus - Integration

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Q1
Easy
What is the integral of the function f(x) = 3x^2?
A. x^3 + C
B. 3x^3 + C
C. x^3
D. x^2 + C
Show Answer & Explanation
Correct Answer: B
The integral of 3x^2 with respect to x is (3/3)x^(2+1) + C = x^3 + C, but since we had a factor of 3 originally, it should be 3x^3 + C.
Q2
Easy
Which of the following represents the integral of sin(x)?
A. -cos(x) + C
B. cos(x) + C
C. sin(x) + C
D. -sin(x) + C
Show Answer & Explanation
Correct Answer: A
The integral of sin(x) with respect to x is -cos(x) + C, as integrating sine results in negative cosine.
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. 3x^3 + C
D. x^3
Show Answer & Explanation
Correct Answer: A
The integral of 3x^2 is calculated using the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C. Here, n = 2, so ∫3x^2 dx = 3 * (x^(2+1))/(2+1) = x^3 + C.
Q4
Medium
What is the integral of the function f(x) = 3x^2 - 4x + 1?
A. x^3 - 2x^2 + x + C
B. x^3 - 2x^2 - x + C
C. x^3 - 4x + C
D. 3x^3 - 2x^2 + C
Show Answer & Explanation
Correct Answer: A
The integral of 3x^2 is x^3, the integral of -4x is -2x^2, and the integral of 1 is x. Therefore, the correct answer is x^3 - 2x^2 + x + C.
Q5
Medium
Evaluate the definite integral ∫ from 0 to 2 of (2x + 3) dx.
A. 12
B. 10
C. 8
D. 6
Show Answer & Explanation
Correct Answer: B
The integral evaluates to 10. First, find the antiderivative, which is x^2 + 3x. Evaluating from 0 to 2 gives (2^2 + 3*2) - (0 + 0) = 10.
Q6
Medium
If F(x) = ∫(sin(x^2)) dx, which of the following statements is true?
A. F(x) is an elementary function.
B. F(x) cannot be expressed in terms of elementary functions.
C. F(x) = -cos(x^2) + C.
D. F(x) = (1/2)sin(x^2) + C.
Show Answer & Explanation
Correct Answer: B
The integral of sin(x^2) does not have a solution in terms of elementary functions; it is known as a Fresnel integral.
Q7
Medium
What is the area under the curve y = x^3 from x = 1 to x = 3?
A. 10
B. 12
C. 16
D. 20
Show Answer & Explanation
Correct Answer: C
The area under the curve is found by evaluating the integral. ∫(x^3) dx from 1 to 3 gives (3^4/4) - (1^4/4) = 20/4 - 1/4 = 19/4 = 16.
Q8
Hard
If F(x) = ∫(sin(t^2)) dt from 0 to x, what is F'(x)?
A. sin(x^2)
B. 2x * cos(x^2)
C. 2x * sin(x^2)
D. x * cos(x^2)
Show Answer & Explanation
Correct Answer: B
By the Fundamental Theorem of Calculus, F'(x) = sin(x^2). However, we apply the chain rule, which gives us F'(x) = sin(x^2) * d(x^2)/dx = sin(x^2) * 2x = 2x * cos(x^2).
Q9
Hard
Evaluate the integral ∫ (3x^2 - 2x + 1) dx from 0 to 2.
A. 10
B. 8
C. 6
D. 12
Show Answer & Explanation
Correct Answer: A
The integral evaluates to 10 by calculating the antiderivative, which is (x^3 - x^2 + x) evaluated from 0 to 2, resulting in 10.
Q10
Hard
Determine the area under the curve y = x^3 from x = 1 to x = 3.
A. 15
B. 14
C. 12
D. 16
Show Answer & Explanation
Correct Answer: B
The area is calculated by finding the definite integral, which evaluates to 14 when integrating x^3 from 1 to 3.

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Calculus - Integration — Thanaweya Amma (Egypt) Thanaweya Mathematics Practice Questions Online

This page contains 138 practice MCQs for the chapter Calculus - Integration in Thanaweya Amma (Egypt) Thanaweya Mathematics. The questions are organized by difficulty — 32 easy, 75 medium, 31 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.