Types of Integrals

Integral are used to solve various types of problems in Calculus, Physics, etc. The integrals are of two types, that are,

• Indefinite Integrals
• Definite Integrals
• Improper Integrals

Indefinite Integrals

Indefinite Integrals are used to find the integrals of the function when the limit of the integration is given. While solving the indefinite integrals we always have the constant of integration in the solution. The integration of the function g(x) is calculated as,

∫ g(x) = G(x) + c

Definite Integrals

Definite Integrals is the integral of the function with the limit of the integration given. Definite integrals gives the value of the function in numerical form. The definite integral of the function is given as,

∫_a^b g(x) dx = G(b) – G(a)

Improper Integrals

Improper integrals arise when the function being integrated is unbounded or has infinite discontinuities within the interval of integration. They are evaluated by considering limits as one or both of the integration limits approach infinity or approach points of discontinuity within the interval.

Here’s a more detailed explanation along with examples:

Infinite Intervals

An improper integral with an infinite interval occurs when one or both of the integration limits are infinite.

• Example: Consider the function f(x)=1/x²​ over the interval [1,∞).
  ∫₁^∞​ 1/x²​ dx
  This integral represents the area under the curve f(x) from x=1 to x=∞. To evaluate it, we compute the limit:
  lim_t → ∞ ∫₁^t​ 1/x²​ dx
  If the limit exists, the integral is said to converge; otherwise, it diverges.

Infinite Discontinuities

An improper integral with infinite discontinuities occurs when the function has a vertical asymptote or an infinite discontinuity within the interval of integration.

• Example: Consider the function g(x)= 1/√x​ over the interval [0,1].
  ∫₀¹ ​ 1/√x​ dx
  The function g(x) has a vertical asymptote at x=0. To evaluate the integral, we compute the limit:
  lim_a → 0⁺ ∫_a¹ ​ 1/√x​ dx
  If the limit exists, the integral converges; otherwise, it diverges.

Functions with Infinite Integrals

Some functions have integrals that extend to infinity due to their behavior.

• Example: The function h(x)=1/x over the interval [0,1].
  ∫₀¹​ 1/x ​dx
  This integral is improper because it becomes infinite at x=0. To evaluate it, we compute the limit:
  lim_a → 0⁺∫_a¹​ 1/x ​dx
  If the limit exists, the integral converges; otherwise, it diverges.

Integrals: An integral in mathematics is a continuous analog of a sum that is used to determine areas, volumes, and their generalizations. Performing integration is the process of computing an integral and is one of the two basic concepts of calculus.

Integral in Calculus is the branch of Mathematics that deals with finding integrals of the given functions. The branch of calculus that deals with integral is called Integral Calculus.

In this article, we will learn about the Integral definition, types, formulas, properties, examples solved problems, etc.