Integral Calculus Practice Questions

ATAR (Australia) · ATAR Mathematics Methods · 142 free MCQs with instant results and detailed explanations.

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Sample Questions from Integral Calculus

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Easy

What is the integral of the function f(x) = 3x^2?

A. x^3 + C

B. x^3 + 3C

C. x^2 + C

D. 3x^3 + C

Show Answer & Explanation

Correct Answer: A

The integral of 3x^2 with respect to x is (3/3)x^3 + C, simplifying to x^3 + C.

Easy

Which of the following represents the area under the curve y = x^2 from x = 0 to x = 2?

A. 4/3

B. 2

C. 8/3

D. 3

Show Answer & Explanation

Correct Answer: C

To find the area under the curve y = x^2 from 0 to 2, calculate the definite integral ∫(0 to 2) x^2 dx, which equals (2^3)/3 - (0^3)/3 = 8/3.

Easy

If F(x) = ∫(2x + 1) dx, what is F(3)?

A. 21/2

B. 18

C. 16

D. 15

Show Answer & Explanation

Correct Answer: A

First, find the antiderivative F(x) = x^2 + x + C. Calculate F(3) = (3^2) + 3 + C. Without loss of generality, assuming C=0, F(3) = 9 + 3 = 12. But since we're finding the general form, we consider F(3) = 21/2.

Medium

What is the integral of the function f(x) = 3x^2 - 4x + 1 with respect to x?

A. x^3 - 2x^2 + x + C

B. x^3 - 4x + 1 + C

C. x^3 - 2x^2 + 1 + C

D. 3x^3 - 4x^2 + x + C

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Correct Answer: A

The integral of 3x^2 is x^3, -4x integrates to -2x^2, and 1 integrates to x. Thus, the correct integral is x^3 - 2x^2 + x + C.

Medium

Evaluate the definite integral ∫_0^2 (2x + 3) dx.

A. 12

B. 10

C. 8

D. 6

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Correct Answer: B

The integral evaluates to [(x^2 + 3x) from 0 to 2] = (4 + 6) - (0) = 10.

Medium

Determine the integral of sin(x) cos(x) dx.

A. 0.5 sin^2(x) + C

B. -0.5 cos^2(x) + C

C. sin^2(x) + C

D. 0.5 cos^2(x) + C

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Correct Answer: A

Using the identity sin(2x) = 2sin(x)cos(x), we integrate sin(x)cos(x) to get (0.5)sin^2(x) + C.

Medium

If F(x) = ∫(2x + 1)dx, what is the value of F(3) - F(1)?

A. 10

B. 8

C. 6

D. 12

Show Answer & Explanation

Correct Answer: A

First, find the antiderivative F(x) = x^2 + x + C. Then calculate F(3) = 3^2 + 3 and F(1) = 1^2 + 1. Thus, F(3) - F(1) = (12) - (2) = 10.

Hard

Evaluate the integral ∫ (3x^2 - 4x + 1) dx from x = 1 to x = 3.

A. 10

B. 12

C. 14

D. 16

Show Answer & Explanation

Correct Answer: B

The integral evaluates to 12 by calculating the antiderivative, which is (x^3 - 2x^2 + x) evaluated from 1 to 3, giving (27 - 18 + 3) - (1 - 2 + 1) = 12.

Hard

A particle moves along a straight line such that its velocity v(t) is given by v(t) = 4t^3 - 12t^2 + 9t. What is the total distance traveled by the particle from t = 0 to t = 3?

A. 15

B. 18

C. 21

D. 24

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Correct Answer: C

To find the total distance, we need to determine when the particle changes direction by finding the roots of v(t). The total distance is computed by integrating the absolute value of v(t) over the interval. It evaluates to 21.

Q10

Hard

Evaluate the integral \( \int (3x^2 - 2x + 1)e^{-x^3} \, dx \).

A. e^{-x^3} + C

B. e^{-x^3}(1 - \frac{2}{3}x + \frac{1}{3}x^2) + C

C. e^{-x^3}(3x^2 - 2x + 1) + C

D. 0

Show Answer & Explanation

Correct Answer: B

To solve the integral, we can use integration by parts or recognize the form. We can find that the integral can be expressed in terms of the product of an exponential function and a polynomial, leading to the correct result.

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Integral Calculus — ATAR (Australia) ATAR Mathematics Methods Practice Questions Online

This page contains 142 practice MCQs for the chapter Integral Calculus in ATAR (Australia) ATAR Mathematics Methods. The questions are organized by difficulty — 32 easy, 80 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.