What are Indefinite Integrals?
Integrals are also known as anti-derivatives. Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-differentiation.
Consider a function f(x) = sin(x), the derivative of this function if f'(x) = cos(x). So, the integration of f'(x) should give back the function f(x). Notice that for every function f(x) = sin (x) + C, the derivative is the same because the constant becomes zero after differentiation. Thus, anti-derivatives are not unique, for every function, its anti-derivatives are infinite.
[Tex]\frac{d}{dx}(sin(x) + C) = cos(x)[/Tex]
This constant C is called an arbitrary constant.
A new symbol is used for denoting integrals [Tex]\int [/Tex]. This will represent the integration operation over any function. The table below represents the symbols and meanings related to integrals.
| Symbol/Term/Meaning | Meaning |
|---|---|
| [Tex]\int f(x)dx[/Tex] | Integral of f with respect to x |
| f(x) in [Tex]\int f(x)dx[/Tex] | Integrand |
| x in [Tex]\int f(x)dx[/Tex] | Variable of integration |
| Integral of f(x) | A function such that F'(x) = f(x) |
There are certain formulas and rules which when kept in mind, help us simplify the calculating and do it fast. The reverse power rule is one of the rules that help us in the integration of polynomials and other functions.
Reverse Power Rule
This rule helps in integrating the functions which have terms of the form xn.
[Tex]\int x^ndx = \frac{x^{n+1}}{n+1} + C[/Tex]
Here, C is the arbitrary constant, and n ≠ 1.
In this rule, the exponent of the variable is increased by 1 and then the result is divided by the new exponent value. The table below gives integrals of some standard functions.
| Function | Integral |
|---|---|
| sin(x) | -cos(x) |
| cos(x) | sin(x) |
| ex | ex |
| sec2(x) | tan(x) |
| [Tex]\frac{1}{x}[/Tex] | ln(x) |
Integration Formulas
Indefinite Integrals: The derivatives have been really useful in almost every aspect of life. They allow for finding the rate of change of a function. Sometimes there are situations where the derivative of a function is available, and the goal is to calculate the actual function whose derivative is given.
In this article, we will discuss Indefinite Integrals, graphical interpretation, formulas, and properties.
Table of Content
- What are Indefinite Integrals?
- Graphical Interpretation of Integrals
- Integrals by Graphs
- Calculating Indefinite Integral
- All Formulas of Indefinite Integrals
- Properties of Indefinite Integrals
- Property of Sum
- Property of Difference
- Property of Constant Multiple
- Difference Between Indefinite Integral and Definite Integral
- Indefinite Integrals Examples
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