# Integral of 1 by x^3

Integral of 1/x^3 along with its formula and proof with examples. Also learn how to calculate integration of 1/x3 with step by step examples.

Alan Walker-

Published on 2023-05-03

## Introduction to the integral of 1/x^3

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to an algebraic function 1/x3. You will also understand how to compute 1 by x^3 integral by using different integration techniques.

## What is the integral of 1/x^3?

The integral of 1/x^3 is an antiderivative of the 1/x^3 function which is equal to the negative of 1/2x^2. It is also known as the reverse derivative of the function 1/x^3 which is an algebraic function. It can be calculated by using the power rule of integral. This rule is written as;

$\int x^n dx=\frac{x^{n+1}}{n+1}+c$

This formula says that the integral of any algebraic function with some exponent, can be calculated by adding 1 in its exponent and dividing by the new exponent i.e n+1.

### Integral of 1 by x^3 formula

The formula of integral of 1 x 3 cube contains integral sign, coefficient of integration and the function as 1/x^3. It is denoted by ∫(1/x^3)dx. In mathematical form, the integral of 1/x^3 is:

$\int \frac{1}{x^3}dx = -\frac{1}{2x^2}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.

## How to calculate the integral of 1/x3?

The integral of 1 by x^3 is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of x^-3 by using:

1. Integration by parts
2. Definite integral

## Integral 1/x3 by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the integral of 1 by x3 by using integration by parts.

### Proof of integral of 1/x3 by using integration by parts

To find the integral 1/(x^3) using integration by parts, we can use the following formula:

$I=f(x)\int g(x)dx-\int[\frac{d}{dx}(f(x))\int g(x)dx]dx$

Suppose that,

$f(x) = \frac{1}{x^3}$