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.

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.

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.

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.

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 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.