Mathematics Class 12 - Integrals Notes

# Integrals

In this chapter, you will learn about integrals, their properties, and how to solve problems using integration. By the end of this chapter, you will be able to evaluate definite and indefinite integrals, understand their applications, and solve related problems in calculus.

## Key Concepts

• Integral: The reverse process of differentiation; used to find areas, volumes, and more.
• Indefinite Integral: Represents a family of functions and includes a constant of integration (C).
• Definite Integral: Represents the area under a curve between two limits.
• Integration: The process of finding an integral.

## Indefinite Integrals

The indefinite integral of a function f(x) is written as:

∫ f(x) dx = F(x) + C

• F(x) is the antiderivative of f(x).
• C is the constant of integration.

Example: ∫ x² dx = (1/3)x³ + C

## Definite Integrals

The definite integral of f(x) from a to b is written as:

∫_a^b f(x) dx = F(b) - F(a)

• a and b are the lower and upper limits of integration.
• F(x) is any antiderivative of f(x).

Example: ∫₀² x dx = [½x²]₀² = 2

## Properties and Applications

• ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
• ∫_a^b f(x) dx = -∫_b^a f(x) dx
• Integration is used to find areas under curves, volumes, and in physics problems.

## Methods of Integration

• Substitution Method
• Integration by Parts
• Partial Fractions

## Practice Questions

1. Find ∫ (3x² + 2x + 1) dx.
2. Evaluate ∫₁³ x dx.
3. If F(x) = ∫ f(x) dx and F(2) = 5, F(0) = 1, find ∫₀² f(x) dx.
4. Find ∫ sin(x) dx.
5. Evaluate ∫₀^π sin(x) dx.

## Challenge Yourself

• Use substitution to solve ∫ 2x cos(x²) dx.
• Find the area under the curve y = x² from x = 0 to x = 2.
• Integrate by parts: ∫ x e^x dx.

## Did You Know?

• The symbol ∫ was introduced by Leibniz and comes from the Latin word "summa" (sum).
• Integration is the reverse process of differentiation.

## Glossary

• Integral: The result of integrating a function.
• Antiderivative: A function whose derivative is the given function.
• Definite Integral: Integral with upper and lower limits; gives a number.
• Indefinite Integral: Integral without limits; gives a family of functions.

## Answers to Practice Questions

1. x³ + x² + x + C
2. [½x²]₁³ = (½ × 9) - (½ × 1) = 4
3. ∫₀² f(x) dx = F(2) - F(0) = 5 - 1 = 4
4. -cos(x) + C
5. [-cos(x)]₀^π = [-cos(π)] - [-cos(0)] = [1 - (-1)] = 2

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Practice integration to master calculus and solve advanced mathematics problems!