# Integrals Notes, Class 12 Mathematics | SwaVid

Integrals, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers.

Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/integrals

Source: https://www.swavid.com/cbse/class-12/maths/chapter/integrals

# Integrals

## Formulas and definitions

## Key ideas

## Common mistakes

## Quick checks

## Keep going

## Questions parents ask

## Class 12 Mathematics on SwaVid

### Integration as an inverse process

### Integration by substitution

### Integrals of trigonometric and rational forms

### Integration by parts

### Fundamental theorem of calculus

### Properties of definite integrals

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SwaVid › CBSE › Class 12 › Mathematics › Integrals

Notes written by SwaVid from the NCERT textbook.

Class 12 Mathematics, Chapter 7. Key ideas, formulas and common mistakes.

CBSE does not print marks for a single chapter.

Short answer. This chapter covers integration as the inverse process of differentiation, including indefinite and definite integrals along with their properties. It provides systematic methods such as substitution, partial fractions, and integration by parts to evaluate standard integrals.

Source: NCERT Mathematics Part II (Reprint 2026-27) , Chapter 7, Integrals

An integral or antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). Indefinite integrals include an arbitrary constant C because the derivative of any constant is zero.

When an integrand contains a function and its derivative, a change of variable transforms the integral into a simpler form. The differential dx is replaced using dt = g'(x) dx.

Integrands involving powers of trigonometric functions are simplified using trigonometric identities. Integrals of rational functions where the denominator factors into linear or quadratic terms are solved using partial fractions.

The integral of a product of two functions is found by taking the first function times the integral of the second minus the integral of the derivative of the first times the integral of the second. The order of functions is chosen using the ILATE rule.

The definite integral ∫ from a to b of f(x) dx represents the net area under the curve y = f(x). The second fundamental theorem states that this equals F(b) - F(a), where F(x) is any antiderivative of f(x).

Standard properties simplify definite integrals, including splitting intervals, shifting variables, and using symmetry for odd and even functions. Replacing x with a + b - x is one of the most widely used properties.

Try each question first. Then tap it to see the answer.

Substitute t = x², so dt = 2x dx.

The integral becomes ∫ cos(t) dt.

Evaluate: ∫ cos(t) dt = sin(t) + C.

Substitute back t = x².

sin(x²) + C

The antiderivative of 1 / (1 + x²) is tan⁻¹(x).

Apply the limits: [tan⁻¹(x)] from 0 to 1 = tan⁻¹(1) - tan⁻¹(0).

Compute values: tan⁻¹(1) = π / 4 and tan⁻¹(0) = 0.

π / 4

Let f(x) = sin⁵(x).

Check symmetry: f(-x) = sin⁵(-x) = (-sin(x))⁵ = -sin⁵(x) = -f(x).

Since f(x) is an odd function, apply the property ∫ from -a to a of f(x) dx = 0.

0

Application of Integrals .

Continuity and Differentiability .

Differential Equations .

The 13 chapters of Class 12 Mathematics .

CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all.

No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed.

Application of Integrals.

SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works.

- Ch 7 NCERT book
- 80 marks, theory paper
- 3 quick checks
- Omitting the constant of integration C in indefinite integrals . Always add + C to the final answer of an indefinite integral, because antiderivatives are unique only up to an additive constant.
- Forgetting to change limits of integration during substitution in definite integrals . When substituting t = g(x), evaluate t at both x = a and x = b to form new limits, or fully back-substitute x before using the original limits.
- Applying partial fractions directly on an improper rational fraction . Check the degrees of the numerator and denominator first. If the numerator degree is greater than or equal to the denominator degree, divide the polynomials before taking partial fractions.
- Missing the outer factor 1 / a in inverse trigonometric integral results . Note that ∫ 1 / (x² + a²) dx gives (1 / a) tan⁻¹(x / a) + C, while ∫ 1 / √(a² - x²) dx gives sin⁻¹(x / a) + C without an outer 1 / a.

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