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Chapter Analysis
Advanced30 pages • EnglishQuick Summary
Chapter 7 of Class 12 Maths Part 2 focuses on integrals, exploring concepts such as indefinite integrals, definite integrals, and the fundamental theorem of calculus. The chapter discusses various methods for solving integrals including substitution, integration by parts, and using partial fractions. It includes exercises on evaluating integrals, the significance of the area function, and the application of definite integrals in real-world problems, providing a comprehensive understanding of integral calculus.
Key Topics
- •Definitions of Indefinite and Definite Integrals
- •Fundamental Theorems of Calculus
- •Integration by Substitution
- •Integration by Parts
- •Integration using Partial Fractions
- •Applications of Definite Integrals
- •Properties and Solutions of Standard Integrals
- •Techniques of Solving Complex Integrals
Learning Objectives
- ✓Understand and apply the fundamental theorems of integral calculus.
- ✓Use substitution and integration by parts to evaluate complex integrals.
- ✓Apply the method of partial fractions to simplify and integrate rational functions.
- ✓Solve real-world applications using definite integrals.
- ✓Recognize and manipulate standard integral forms for complex problems.
- ✓Develop strategies to solve definite integrals with variable bounds.
Questions in Chapter
Evaluate the integral \(\int (2x^3 - 3x^2 + x - 5)\, dx\).
Page 271
Evaluate \(\int_{0}^{\pi/2} \sin x \, dx\).
Page 280
Find the area under the curve \(y = x^2\) from \(x = 0\) to \(x = 3\).
Page 285
Additional Practice Questions
Solve \(\int (x^2 + 4x + 4)\, dx\).
mediumAnswer: The integral of \(x^2 + 4x + 4\) with respect to x is \(\frac{x^3}{3} + 2x^2 + 4x + C\).
Evaluate \(\int e^{2x} \cos(3x) \, dx\).
hardAnswer: Use integration by parts and the fact that \(\int e^{ax}\cos(bx)\,dx\) involves repeated integration by parts involving trigonometric identities.
Determine \(\int \frac{1}{\sqrt{x^2 + 9}} \, dx\).
mediumAnswer: Set \(x = 3\tan\theta\), dx becomes \(3\sec^2\theta d\theta\), and solve the resulting integral.
Calculate \(\int_{1}^{2} (3x^2 - 2x + 1) \, dx\).
easyAnswer: Calculate the antiderivative, evaluate at the upper and lower limits, giving \(\frac{8}{3} - \frac{5}{3} = 1\).
Integrate \(\int \ln x \, dx\).
mediumAnswer: Using integration by parts, set \(u = \ln x\) and \(dv = dx\), yielding \(x \ln x - x + C\).