Class 12 Maths - KERALA

Integrals

The Chapter Integrals in Class 12 Mathematics for Kerala SCERT explores integration as the inverse process of differentiation and as the area under a curve. Students will study various methods of integration including substitution, partial fractions, and integration by parts. The chapter also covers definite integrals and the fundamental theorem of calculus. This is a high-scoring and crucial chapter for the Kerala Board examinations, frequently appearing in both short answer and long essay questions, forming the foundation for applications of integrals in finding areas.

Start Learning Free

Key Concepts

Indefinite Integrals

The anti-derivative of a function represented generally with an arbitrary constant C, denoted as the integral of f(x) dx.

Integration by Substitution

A technique used to transform a complicated integral into a simpler one by substituting a variable with a function of another variable.

Partial Fractions

A method to break down a complex rational function into simpler fractions that can be easily integrated individually.

Integration by Parts

A rule based on the product rule of differentiation used to integrate the product of two functions, using the formula involving u and v.

Definite Integrals

An integral with lower and upper limits, representing the exact net area under a curve between two points on the x-axis.

Properties of Definite Integrals

Special algebraic rules regarding limits and symmetry (such as integrating from -a to a) that simplify the evaluation of definite integrals.

Important Formulas

Integral of x^n dx = (x^(n+1))/(n+1) + C (where n != -1)

Integral of (1/x) dx = log|x| + C

Integral of e^x dx = e^x + C

Integral of sin(x) dx = -cos(x) + C

Integral of cos(x) dx = sin(x) + C

Integral of sec^2(x) dx = tan(x) + C

Integration by parts: Integral of (u * v) dx = u * Integral(v dx) - Integral((du/dx) * Integral(v dx)) dx

Definite integral fundamental theorem: Integral from a to b of f(x) dx = F(b) - F(a)

Board Exam Info

In the Kerala SCERT Class 12 Mathematics board examination, Integrals carries a significant weightage of around 12 to 15 marks. Common question types include direct evaluation of indefinite integrals using substitution and partial fractions, multi-step integration by parts, and proving properties of definite integrals.

Frequently Asked Questions

Why do we add '+ C' in indefinite integrals?

We add the constant C because differentiation of a constant is zero, meaning there could be infinite functions whose derivative matches the given integrand.

How do I choose between substitution and integration by parts?

Use substitution when you see a function and its derivative multiplied together in the integrand. Use integration by parts when you have a product of two completely different types of functions, like an algebraic function and a trigonometric or logarithmic function.

Are properties of definite integrals important for exams?

Yes, absolutely. Properties like the King Property (Integral from a to b of f(x) dx = Integral from a to b of f(a+b-x) dx) are frequently tested in 4-mark or 6-mark questions in the Kerala board exams.

Learn Integrals with Your AI Tutor

10 different ways to study this chapter. Free for 3 chapters per day.

Lecture

Key Points

Interactive

Quiz

Flashcards

Start Learning Free

More Maths Chapters - KERALA Class 12

Relations And Functions Inverse Trigonometric Functions Matrices Determinants Continuity And Differentiability Application Of Derivatives Application Of Integrals Differential Equations Vector Algebra Three Dimensional Geometry Linear Programming Probability