# Integration

# Integration

## Introduction

**Integration**is the reverse process of differentiation, finding the original function when given its derivative.- It is used for various applications in mathematics, including finding areas under curves or solving differential equations.

## Definite and Indefinite Integration

**Indefinite integration**, also referred to as anti-differentiation, involves finding a function F(x) which has the derivative f(x).**Definite integration**involves finding the exact value of the area under the curve from one point to another, represented by limits.

## Techniques of Integration

**Integration by substitution**involves changing the variable in an integral to make it easier to integrate.**Integration by parts**, derived from the product rule of differentiation, is used when integrating the product of two functions.**Integration using partial fractions**simplifies the function into simpler fractions which are easier to integrate individually.

## Numerical Integration

- Numerical methods for integration like
**Trapezium Rule**,**Simpson’s Rule**, and the**Mid-Ordinate Rule**are used when a function is too complex to integrate analytically or only numerical data is available. **Trapezium Rule**approximates the area under the curve by splitting it into several trapezoids of equal width.**Simpson’s Rule**provides a more accurate approximation by using parabolas to approximate each segment of the curve.**Mid-Ordinate Rule**uses rectangles under the curve at midpoints to approximate the area.

## Error and Accuracy in Numerical Integration

- The accuracy of numerical integration depends on the number and width of the subdivisions; more and narrower subdivisions generally yield more accurate results.
- Errors in numerical integration can be reduced by using
**more precise floating point numbers**,**increasing the number of intervals**, or**combining different methods**.

## Applications of Integration

- Integration is used in many areas of mathematics and physics, such as calculating
**areas and volumes**, solving**differential equations**, and determining the**centroid**of a region. - Integration also plays a key role in statistics, where it is used to calculate
**probability distributions**and**cumulative distribution functions**.