Are these practice questions for Integrals based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Integrals free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Integrals mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Integrals for full coverage.

How many practice questions are there for Integrals?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Integrals Class 12 Mathematics Practice Questions with Answers

easy

Evaluate the integral using a suitable substitution:

\int \frac{\cos x}{1 + \sin x} dx

Try solving it in your notebook first, then check the solution.

easy

Evaluate the definite integral:

\int_{0}^{1} \frac{dx}{\sqrt{4-x^2}}

Try solving it in your notebook first, then check the solution.

easy

Calculate the integral:

\int \frac{x^2+4}{x^2+1} dx

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easy

Apply integration by parts to solve:

\int x \sec^2 x dx

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easy

Recall the standard integral of

\frac{1}{x}

.

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easy

Identify the 'integrand' and the 'variable of integration' in the expression

\int (3t^2 + \cos t) dt

.

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easy

Define an anti derivative or primitive of a function.

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easy

Justify why

\int_{-a}^{a} f(x) dx = 0

if

f(x)

is an odd function, by interpreting the definite integral as a net signed area.

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easy

State the power rule for integration.

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easy

Calculate the indefinite integral:

\int (5x^4 - 2\sec^2 x + \frac{3}{x}) dx

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medium

State the formula for integration by parts and identify which function is typically chosen as the 'first function'.

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medium

A student is asked to evaluate

\int \frac{dx}{x(x^5+1)}

. They attempt to use partial fractions as

\frac{A}{x} + \frac{Bx^4+Cx^3+Dx^2+Ex+F}{x^5+1}

. Critique this approach. Propose a more efficient method and use it to find the correct integral.

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medium

Analyze and evaluate the integral

\int_{-\pi/2}^{\pi/2} x^3 \cos x dx

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medium

Demonstrate the use of integration by parts to find the integral of

\log x

.

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medium

Evaluate:

\int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx

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medium

Solve the integral:

\int \frac{3x+1}{(x+1)(x-2)} dx

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medium

Formulate the integral for finding the anti-derivative of

\sin^{-1}\left(\frac{2x}{1+x^2}\right)

and propose a method to solve it by first simplifying the integrand.

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medium

Derive the formula for

\int \sqrt{a^2 - x^2} dx

using the method of integration by parts.

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medium

Calculate the definite integral:

\int_{0}^{\pi/4} \tan^2 x dx

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medium

Formulate an indefinite integral of the form

\int e^x [f(x) + f'(x)] dx

where

f(x) = \frac{1}{x^2}

, and then state its solution.

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medium

Explain what the 'constant of integration' represents in an indefinite integral.

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medium

Find the integral:

\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx

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medium

List the standard integrals for the following six special rational and irrational functions: (i)

\int \frac{dx}{x^2 - a^2}

(ii)

\int \frac{dx}{a^2 - x^2}

(iii)

\int \frac{dx}{x^2 + a^2}

(iv)

\int \frac{dx}{\sqrt{x^2 - a^2}}

(v)

\int \frac{dx}{\sqrt{a^2 - x^2}}

(vi)

\int \frac{dx}{\sqrt{x^2 + a^2}}

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medium

Evaluate the integral

\int e^{2x} \cos(3x) dx

.

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medium

A student evaluates the integral

\int_{-1}^{1} \frac{1}{x^2} dx

as

[\frac{-1}{x}]_{-1}^{1} = (\frac{-1}{1}) - (\frac{-1}{-1}) = -1 - 1 = -2

. Critique this evaluation and justify your conclusion.

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medium

List the standard forms of partial fractions for the rational functions (i)

\frac{px^2+qx+r}{(x-a)(x-b)(x-c)}

and (ii)

\frac{px^2+qx+r}{(x-a)^2(x-b)}

.

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medium

First, prove the property

\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx

. Then, use this property to design a solution for evaluating

\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\cot x}}

.

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medium

When integrating

\int x \tan^{-1}(x) dx

by parts, justify the choice of

\tan^{-1}(x)

as the first function.

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medium

State the Second Fundamental Theorem of Integral Calculus.

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medium

List the formulas for the integrals of the three special functions involving square roots.

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medium

Find the integral of

e^x(\sin x + \cos x)

.

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medium

State the property of a definite integral when its upper and lower limits are the same.

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medium

Calculate:

\int \frac{dx}{x^2 - 6x + 5}

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medium

Recall the standard integrals for

\int \sec x \, dx

and

\int \operatorname{cosec} x \, dx

.

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medium

When evaluating an integral of the form

\int \frac{\cos x}{\cos x + \sin x} dx

, a common strategy is to express the numerator as

\frac{1}{2}[(\cos x + \sin x) + (\cos x - \sin x)]

. Justify why this specific algebraic manipulation is an effective strategy for solving the integral.

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hard

Evaluate the definite integral

\int_{0}^{1} \cot^{-1}(1-x+x^2) dx

.

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hard

Evaluate the integral

\int \frac{x^2+4}{x^4+16} dx

.

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hard

Evaluate the definite integral

\int_{0}^{\pi} \frac{x \sin x}{1 + \sin x} dx

. Justify the key property used in your solution.

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hard

Explain the method of 'Integration by Substitution'. Describe the key steps involved in this process for a definite integral.

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hard

Calculate the definite integral:

\int_{1}^{2} \frac{5x^2}{x^2+4x+3} dx

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hard

Derive the reduction formula for

I_n = \int \sec^n x dx

, where

n

is a positive integer greater than 2. Use the derived formula to evaluate

\int \sec^4 x dx

.

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hard

Describe the property

\mathbf{P_7}

of definite integrals concerning even and odd functions. Explain why this property is useful.

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hard

Without performing explicit integration, evaluate

\int_{0}^{2\pi} \sin^7(x) dx

and justify your reasoning using the properties of definite integrals.

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hard

Solve the integral:

\int \frac{x^2}{(x^2+1)(x^2+4)} dx

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hard

State the property

\mathbf{P_4}

of definite integrals and explain what it signifies.

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IntegralsClass 12 Mathematics Practice Questions