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.