Integration is an important part of calculus. It involves finding the anti-derivative of a function and is used to solve integrals. Integration has numerous applications in various fields, such as mathematics, physics, and engineering.

This article serves as a comprehensive guide to integration, covering everything from integration formulas to methods for finding integrals. It also explains the properties and real-world applications of integration through solved examples. Let’s start exploring the topic of Integration.

What is Integration?

The process of determining the function from its derivative is called Integration. In other words, the procedure of finding the anti-derivatives of the function is called the integration. The result obtained after the integration is called integral. The integration can be done using multiple methods like integration by substitution, integration by parts, integration by partial fraction, etc.

Integration Definition

If f is the positive continuous function defined over an interval [a, b] then, the area between the function f graph and x-axis results in the integration of f w.r.t x. The area under the curve gives the definite integration of f.

Integration Symbol

The symbol of integration is ∫. For the definite integral we apply limits and use symbol ∫^a_b .

Rules for Integration

Some important rules of integration are:

• Power Rule
• Addition Rule
• Subtraction Rule
• Constant Multiple Rule 

Power Rule of Integration

The Power Rule of Integration is a fundamental rule for finding the antiderivative (indefinite integral) of a function in the form of a power of x, The power rule of integration is stated as:

∫xⁿ dx = xⁿ⁺¹/ (n + 1)

Addition Rule of Integration

The Addition Rule of Integration allows you to integrate the sum of two functions by integrating each function separately and then adding the results. The addition rule of integration is stated as:

∫{f(x) + g(x)} dx = ∫f(x) dx + ∫g(x) dx

Subtraction Rule of Integration

The Subtraction Rule of Integration is similar to the Addition Rule of Integration. It allows you to integrate the difference of two functions by integrating each function separately and then subtracting the results. The subtraction rule of integration is stated as:

∫{f(x) – g(x)} dx = ∫f(x) dx – ∫g(x) dx

Constant Multiple Rule of Integration

The Constant Multiple Rule of Integration states that if you have a constant multiplied by a function, you can factor out the constant and then integrate the function The multiplication of constant rule of integration can be stated as:

∫k f(x) dx = k ∫f(x) dx,

Where k is constant.

Antiderivative: Integration as Inverse Process of Differentiation

The process of finding the antiderivative i.e., the inverse of the derivative is called integration. If Φ(x) is a function and the derivative of Φ(x) is f(x) then, integration of f(x) results in Φ(x).

(d / dx) {Φ(x)} = f(x) ⇔ ∫f(x) dx = Φ(x) + C

OR

∫[d/dx]g(x) dx = g(x)

Integration Formulas

The various integration formulas are:

1. [Tex]\frac{d}{dx}   [/Tex]{Φ(x)} = f(x) ⇔ ∫f(x) dx = Φ(x) + C
2. ∫xⁿ dx = xⁿ⁺¹/ (n + 1) + C
3. ∫(1 / x) dx = log_e|x| + C
4. ∫e^x dx = e^x + C
5. ∫a^x dx = [a^x/ log_ea] + C
6. ∫sin x dx = -cos x + C
7. ∫cos x dx = sin x + C
8. ∫sec²x dx = tan x + C
9. ∫cosec²x dx = -cot x + C
10. ∫sec x tan x dx = sec x + C
11. ∫cosec x cot x dx = – cosec x + C
12. ∫tan x dx = ln |sec x| + C = -ln |cos x| + C
13. ∫cot x dx = ln |sin x| + C
14. ∫sec x dx = ln |sec x + tan x| + C
15. ∫cosec x dx = ln |cosec x – cot x| + C
16. ∫[Tex]\frac{1}{\sqrt{a^2 -x^2}}   [/Tex]dx = sin^-1(x/a) +C
17. ∫- [Tex]\frac{1}{\sqrt{a^2 -x^2}}   [/Tex]dx = cos^-1(x/a) + C
18. ∫ [Tex]\frac{1}{{a^2 + x^2}}   [/Tex]dx = (1/a) tan^-1 (x/a) + C
19. ∫- [Tex]\frac{1}{{a^2 + x^2}}   [/Tex]dx = (1/a) cot^-1(x/a) + C
20. ∫[Tex]\frac{1}{x\sqrt{x^2 -a^2}}   [/Tex]dx = (1/a) sec^-1(x/a) + C
21. ∫- [Tex]\frac{1}{x\sqrt{x^2 -a^2}}   [/Tex]dx = (1/a)cosec^-1(x/a) + C

Learn more about Integration Formulas.

Types of Integration

Integration can be classified into three types:

• Definite Integration
• Indefinite Integration
• Improper Integration

Definite Integration

The integration with limits is called as the definite integration. The antiderivative p(x) of a continuous function f(x) on the interval [a, b] is known as definite integration. It is represented by and its value equals to p(b) – p(a) where p(b) is antiderivative at x = b and p(a) is the antiderivative at x = a. The a and b are called the limits of integration where a is the lower limit and b is the upper limit of integration. The interval [a, b] is called the interval of the integration. In the definite integration, the constant of integration is not required.

Learn more about Definite Integration.

Indefinite Integration

The integration with no limits is called the indefinite integration. In the indefinite integration, we add a constant with the result called the constant of integration. The integration of f(x) is given by:

∫f(x) dx = P(x) + C

Where,

• x is the variable of integration,
• f(x) is integrand,
• P(x) is antiderivative of f(x), and
• C is constant of Integration.

Improper Integration

The integration whose integrand is not bounded, or the limit of the integral is infinity, then the integration is called as improper integration. Some examples of improper integrations are: [Tex]\int\limits_1^\infty   [/Tex]f(x)dx or [Tex]\int\limits_0^1   [/Tex](dx / x)

Integration Techniques

There are various methods to find the integration of a function. Some of these are listed below:

• Integration by Substitution
• Integration by Parts
• Integration by Partial Fraction

Read more about Method of Integration.

Integration of Basic Functions

There are different integration formulas for different functions. Below we will discuss the integration of different functions in depth and get complete knowledge about the integration formulas.

Integration of Constant Function

The integration of a constant function is given by:

∫k dx = kx + C, where k is constant

Integration of Trigonometric Functions

The integration of trigonometric functions is given by:

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫sec²x dx = tan x + C
• ∫cosec²x dx = -cot x + C
• ∫sec x tan x dx = sec x + C
• ∫cosec x cot x dx = – cosec x + C
• ∫tan x dx = -log |cos x| + C
• ∫cot x dx = log |sin x| + C
• ∫sec x dx = log |sec x + tan x| + C
• ∫cosec x dx = log |cosec x – cot x| + C

Integration of Exponential and Logarithmic Functions

The integration of exponential and logarithmic function is given by:

• ∫(1 / x) dx = log_e|x| + C
• ∫e^x dx = e^x + C
• ∫a^x dx = [a^x/ log_ea] + C

Applications of Integration

There are various applications of integration. Some of them are listed below:

• Integration is used to find area under the curve.
• It is also used to find the volumes.
• Integration is used to find area between the two curves.
• It is used in multiple formulas in physics.

Integration in Physics and Engineering

Integration is used widely in Physics and Engineering.

• In Physics integration is used in multiple formulas for example finding the velocity from acceleration and many more.
• In Engineering integration is used in different fields and in many formulas of engineering mechanics etc.

Integration in Economics and Finance

Integration is also helpful in Economics and Finance to calculate marginal and total revenue, costs, profits, consumer and producer surplus, capital accumulation over a specified time and in the Lorenz curve and Gini coefficient.

Integration vs Differentiation

The basic difference between integration and differentiation is tabulated below: