NCERT Solutions for Class 12 Mathematics Chapter 7 Integrals
NCERT Solutions for Class 12 Mathematics Chapter 7 Integrals covers indefinite integrals as the reverse process of differentiation, key methods of integration such as substitution, partial fractions, and integration by parts, and the evaluation of definite integrals.
Overview of Topics & Subtopics (Click to Jump)
1. Indefinite Integrals and Basic Formulas
Short Note:
• Integration is the reverse process of differentiation; the indefinite integral of a function f(x) is written as ∫f(x)dx = F(x) + C, where F'(x) = f(x).
• Standard formulas include ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫(1/x)dx = ln|x| + C, and ∫eₓdx = eₓ + C.
2. Methods of Integration
Short Note:
• Integration by substitution simplifies an integral by changing the variable, often converting a complex expression into a standard form.
• Integration by parts, given by ∫u dv = uv - ∫v du, is used for products of functions, while partial fractions are used for rational functions with factorable denominators.
3. Definite Integrals
Short Note:
• A definite integral ∫[a to b] f(x)dx represents the net signed area under the curve y = f(x) between x = a and x = b, and is evaluated using the Fundamental Theorem of Calculus.
• Definite integrals have useful properties, such as ∫[a to b] f(x)dx = -∫[b to a] f(x)dx, which simplify calculations in many problems.
NCERT Textbook Exercise Questions
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