Definite Integration Practice Questions

JEE · Mathematics · 1388 free MCQs with instant results and detailed explanations.

1388

Total

441

Easy

603

Medium

344

Hard

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Topics in Definite Integration

Reduction Formulas 251

Leibnitz's Rule 122

Gamma Function 379

Properties of Definite Integrals 275

Walli's Formula 361

Sample Questions from Definite Integration

Here are 10 sample questions. Start a quiz to get randomized questions with scoring.

Easy

Using the property of definite integrals, what is ∫(from -a to a) f(x) dx if f(x) is an odd function?

A. 0

B. a

C. 2a

D. undefined

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Correct Answer: A

For odd functions, the integral over symmetric limits around zero is zero.

Easy

What is ∫(from 0 to a) (x^2) dx?

A. a^3/3

B. a^2/2

C. a^4/4

D. a^5/5

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Correct Answer: A

The integral of x^2 from 0 to a is a^3/3, from the power rule of integration.

Easy

Using the property of integrals, what is ∫(from a to b) f(x) dx + ∫(from b to a) f(x) dx?

A. 0

B. f(a) + f(b)

C. 2∫(from a to b) f(x) dx

D. f(b) - f(a)

Show Answer & Explanation

Correct Answer: A

This property reflects that integrating over a closed interval yields zero.

Medium

Which of the following properties of definite integrals is incorrect?

A. If a < b, then ∫[a to b] f(x) dx = -∫[b to a] f(x) dx

B. ∫[a to b] (f(x) + g(x)) dx = ∫[a to b] f(x) dx + ∫[a to b] g(x) dx

C. ∫[a to b] c*f(x) dx = c*∫[a to b] f(x) dx for any constant c

D. ∫[a to b] f(x) dx = ∫[c to d] f(x) dx for any c, d in the interval [a, b]

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Correct Answer: D

The property ∫[a to b] f(x) dx is only equal to ∫[c to d] f(x) dx if c and d correspond to a transformation of variables. Otherwise, this equality does not hold for arbitrary c and d.

Medium

If f(x) is an odd function, what is the value of ∫[-a to a] f(x) dx?

A. 0

B. a

C. 2a

D. undefined

Show Answer & Explanation

Correct Answer: A

The integral of an odd function over a symmetric interval about the origin always equals zero because the positive and negative areas cancel each other out.

Medium

If \( I = \int_{a}^{b} f(x) \, dx \) and \( J = \int_{a}^{b} f(a + b - x) \, dx \), which of the following statements is true about \( I \) and \( J \)?

A. \( I = J \)

B. \( I = -J \)

C. \( I + J = b - a \)

D. \( I - J = f(a) + f(b) \)

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Correct Answer: A

The integral of a function and its reflection about the midpoint results in the same area under the curve.

Medium

For the integral \( \int_{0}^{1} e^{-x^2} \, dx \), which approximates the area under the curve most accurately?

A. \( \frac{1}{2} \)

B. \( \frac{1}{\sqrt{e}} \)

C. \( \frac{1}{e} \)

D. \( \frac{1}{3} \)

Show Answer & Explanation

Correct Answer: B

The integral does not have a closed form, but can be approximated and evaluated using numerical methods.

Hard

If \( I = \int_{0}^{\pi} \sin^2(x) \, dx \), what is the relationship between \( I \) and the integral \( J = \int_{0}^{\pi} \cos^2(x) \, dx \)?

A. I = J

B. I = 2J

C. I + J = \frac{\pi}{2}

D. I + J = \pi

Show Answer & Explanation

Correct Answer: C

The integrals of \( \sin^2(x) \) and \( \cos^2(x) \) over \( [0, \pi] \) are equal and each contributes equally to half the area under the curve.

Hard

Determine the value of the integral \( I = \int_{-1}^{1} (x^3 - 2x) \, dx \).

A. 0

B. -1

C. 1

D. 2

Show Answer & Explanation

Correct Answer: A

The integrand is an odd function, and the integral of any odd function over a symmetric interval around the origin is zero.

Q10

Hard

If I = ∫_0^π/2 sin^2(x) dx, what is the value of I?

A. π/4

B. π/8

C. π/2

D. π/16

Show Answer & Explanation

Correct Answer: A

Using the identity sin^2(x) = (1 - cos(2x))/2, the integral simplifies to π/4.

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Definite Integration — JEE Mathematics Practice Questions Online

This page contains 1388 practice MCQs for the chapter Definite Integration in JEE Mathematics. The questions are organized by difficulty — 441 easy, 603 medium, 344 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. This chapter covers 5 topics, giving you comprehensive coverage of the entire chapter.