Class 12 Maths - CBSE

Integrals

The chapter 'Integrals' in Class 12 Mathematics explores the reverse process of differentiation, known as anti-derivatives or integration. Students learn indefinite and definite integrals, along with powerful evaluation techniques including substitution, partial fractions, and integration by parts. The Fundamental Theorem of Calculus bridges derivatives and definite integrals. Mastering this chapter is crucial for board exams, carrying a heavy weightage in calculus alongside Continuity and Differentiability and Differential Equations. It is heavily tested through standard evaluation problems and word problems based on finding areas under curves.

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Key Concepts

Indefinite Integrals

The family of all anti-derivatives of a function, represented with an arbitrary constant 'C' because differentiation of a constant is zero.

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 whose derivative is present in the integrand.

Integration by Partial Fractions

A method to break down complex rational algebraic functions 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 integrals of derivatives.

Definite Integrals

An integral with upper and lower limits, representing the net area bounded by the curve, the x-axis, and the ordinates x = a and x = b.

Properties of Definite Integrals

Special rules like the king property that simplify the evaluation of definite integrals by altering limits or combining symmetric functions.

Important Formulas

Int (x^n) dx = (x^(n+1)) / (n+1) + C, where n != -1

Int (1/x) dx = log |x| + C

Int (e^x) dx = e^x + C

Int (sin x) dx = -cos x + C

Int (cos x) dx = sin x + C

Int u * v dx = u * Int v dx - Int (du/dx * Int v dx) dx

Int [f(x) + g(x)] dx = Int f(x) dx + Int g(x) dx

Int_a^b f(x) dx = F(b) - F(a)

Board Exam Info

In the CBSE Class 12 Mathematics board exam, Integrals along with Applications of Integrals typically carry around 10 to 12 marks. Questions frequently include evaluating indefinite integrals using substitution and partial fractions, applying integration by parts, and solving tricky definite integral problems using properties.

Frequently Asked Questions

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

We add '+ C' (the constant of integration) because the derivative of any constant is zero. When reversing differentiation, the original function could have had any constant value, so '+ C' represents all possible answers.

How do I know whether to use substitution or partial fractions?

Use substitution when you see both a function and its derivative multiplied together in the integrand. Use partial fractions when the integrand is a proper rational function with a denominator that can be factored.

How can I remember the ILATE rule for integration by parts?

ILATE helps you choose the first function (u) in integration by parts in the order: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, and Exponential functions.

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