# Integration as Inverse of Differentiation

# Integration as Inverse of Differentiation

## Basic Concepts

**Integration**is essentially the reverse process of differentiation.- Integration and differentiation are
**inverse operations**; one undoes the other. - When we differentiate a function, we get its rate of change. When we integrate a function, we find the original quantity.

## Indefinite Integral

- The
**indefinite integral**, ∫f(x) dx, represents a family of functions, F(x) + C, where C is an arbitrary constant, known as the**constant of integration**. - The function F(x) is an antiderivative or primitive of f(x), meaning F’(x) = f(x).
- Common functions and their antiderivatives include: ∫x^n dx = (1/(n+1))x^(n+1) + C (for n ≠ -1), ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C.

## Definite Integral

- The
**definite integral**, ∫f(x) dx from a to b evaluates to F(b) - F(a), where F’(x) = f(x), in a process known as the**Fundamental Theorem of Calculus**. - This definite integral represents the
**net area**between the function f(x) and the x-axis, from x = a to x = b.

## Applications of Integration

- The process of integration can be used to find
**areas**,**volumes**,**central points**and many other things. - It also plays a fundamental role in solving
**differential equations**, which involve both derivatives and integrals of functions.

## Techniques of Integration

**Substitution**: This technique involves making a substitution to simplify the process of integration.**Integration by parts**: This method is used when the integrand is the product of an ‘easily integrable’ function and another function whose derivative is simpler than the original.**Partial fractions**: This technique is used to simplify the integrand into simpler fraction(s), making it easier to integrate.**Trigonometric substitutions**: These are used when the integrand involves quadratic expressions under a square root or involves expressions such as sin^2(x) or cos^2(x).

## Improper Integrals

**Improper Integrals**are those involving functions that are not defined at one or more points in the interval of integration or the interval of integration is infinite.- To evaluate these integrals, we often use the limit concept, substituting the point of discontinuity or infinity with a variable, calculating the integral, and then taking the limit as that variable approaches the discontinuity or infinity.