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Integrals

Home / Boards / CBSE / Important Questions / Class 12 / Maths / Integrals

Class 12 Maths Chapter 7
Integrals
Important Questions

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For important questions from Class 12 Maths, Chapter 7 Integrals, you can focus on various key topics that are typically covered in this chapter.The specific questions can vary based on your textbook or curriculum.Basic Integration Formulas,Integration by Substitution,For more specific questions, it would be helpful to refer cbse class 12 maths integrals important questions to your textbook or study materials.additional academic resources can be very helpful.Tackling these questions not only boosts confidence but also understanding the concepts.

Introduction

An introduction to integrals in mathematics encompasses several key concepts and principles. Integrals are a fundamental part of calculus, which is a branch of mathematics focused on change and motion. They are primarily use to find areas, volumes, central points, and many useful things.When integrals cannot be solved analytically, numerical methods like the Trapezoidal Rule or Simpson’s Rule are used to estimate their values.Integration is a powerful tool in calculus for cbse class 12 integrals offering ways to solve complex real-world problems. Its applications span across various scientific and engineering disciplines, making it an essential area of study in higher mathematics.

What are Integrals?

Integrals are a fundamental concept in calculus, a branch of mathematics that deals with continuous change. They are used to accumulate quantities and compute areas, volumes, and other values associated with functions. Fundamental theorem of calculus connects the concept of differentiation (finding rates of change) with integration (accumulating quantities). It states that the process of taking an antiderivative (integration) is the reverse of taking a derivative.

Class 12 Integrals Important Questions and Answers

Q1. \text{If} ∫ 2^x dx = f(x) + C, then f(x) is
Options
(a) 2^x \\[4.5 bp] (b)\space 2^x log_e2\\[4.5 bp] (c)\space \dfrac{2^x}{log_e2}\\[4.5 bp] (d)\space \dfrac{2^{x+1}}{x+1}

Ans. (c) \dfrac{2x}{log_e2}\\[2.5 bp] Explanation:
We know that differentiation is the inverse process of integration.
\\ \left(\dfrac{d}{dx}\right)\left(\dfrac{2x}{\text{log}_e2}\right) = \left(\dfrac{1}{log_e2}\right).2^x. \text{log}_e2 = 2^x.

Q2: ∫ cot²x dx equals to.
Options
(a) cot x – x + C
(b) -cot x – x + C
(c) cot x + x + C
(d) -cot x + x + C

Ans. (b) -cot x – x + C

Explanation:
We know that \cot^2 x = \cosec^2x – 1\\[4.5 bp] ∫\cot²x \space dx = ∫ (\cosec^2x – 1) dx = -\cot x - x + C. [\text{Since, }∫ \cosec^2x \text{ dx = -} \cot x + c]\\[2.5 bp] Hence, the correct answer is option (b) -cot x – x + C.

Q3. \int\limits^{\pi}_0 \text{sin}^2 \text{x dx}
Options
(a)\dfrac{π}{2} \\[4.5 bp] (b)\dfrac{π}{4}\\[4.5 bp] (c) 2π
(d) 4π

Ans: (a)\dfrac{π}{2} \\[2.5 bp] Explanation:
\int\limits_0^{\pi} \text{sin 2x dx =} \left( \dfrac{1}{2}\right) \int\limits_0^\pi (1 - \text{cos 2x})\text{ dx}\\ Now, integrate the function and apply the limits, we get
\\ \int\limits_0^{\pi} \text{sin}^2 \text{x dx = } \left( \dfrac{1}{2}\right)(π-0) =\dfrac{π}{2}.\\ Hence, option (a) \dfrac{π}{2} is the correct answer.

Q4. ∫ sec²(7 – 4x)dx = a tan (7 – 4x) + C, what is the value of a?

Explanation:
∫ \sec²(7 – 4x)dx = a \tan (7 – 4x) + C\\[4.5 bp] ∫ \sec²(7 – 4x)dx = \dfrac{[tan (7 - 4x)]}{-4} + C\\[4.5 bp] ∫ \sec²(7 – 4x)dx = \left(-\dfrac{1}{4}\right) \tan (7-4x) + C\\[4.5 bp] \text{Hence, the value of a is -}\dfrac{1}{4}.

Q5: Evaluate
\int\limits_0^2 \space \text{x}^2 \text{dx}

Explanation:
\int\limits_0^2 \text{x}^2 \text{dx} = \left[\dfrac{x^3}{3}\right]^2_0 \\ Now, apply the limits, we get
\\[2.5 bp] \int\limits_0^2 \text{x}^2 \text{dx} = \left(\dfrac{2^3}{3}\right) - 0 = \dfrac{8}{3}

CBSE Class 12 Maths Chapter wise Important Questions

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Conclusion

Integrals are a key component of calculus that have deep theoretical importance and practical applications across a vast range of disciplines. Exploring cbse class 12 maths Integrals important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics.. To enhance understanding, platforms like Oswal.io provide a wealth of resources, including question-answer sets, comprehensive class 12 questions and answers.Utilising these resources ensures students acquire the knowledge necessary to excel in their studies and examinations.

Frequently Asked Questions

Ans: An integral is a mathematical concept used to find areas under curves, sums of infinite series, and many other concepts in calculus. It’s a way of adding slices to find a whole.
Ans: A definite integral calculates the area under a curve between two limits and has a numerical value. An indefinite integral represents a family of functions and includes an arbitrary constant, representing the antiderivative of a function.
Ans: Integrals can be calculated using various methods like substitution, integration by parts, and partial fractions. For some functions, integrals are looked up in a table or computed using software tools.
Ans: This theorem links the concept of differentiation with integration. It states that the integral of a function over an interval can be calculated using its antiderivative at the endpoints of the interval.
Ans: Not all functions have an elementary antiderivative. In such cases, integrals may be computed numerically or represented as special functions.
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