# 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.