Integrals are essential in various fields such as physics, engineering, economics, and statistics. They help solve problems related to motion, area, volume, and total accumulation, making them a powerful tool for understanding and modelling continuous change.

Integral of 1/x² is -(1/x) + C

In this article, we will solve the integral of 1/x².

What is an Integral?

Integrals are of two types that are added below:

Indefinite Integrals

An indefinite integral is essentially the reverse process of differentiation. It involves finding a function whose derivative is a given function. The result of an indefinite integral is a family of functions, often written with a “+ C” at the end to represent an arbitrary constant, since differentiation removes this constant.

Definite Integrals

A definite integral computes the collection of a quantity over an interval. It represents the signed area under the curve of a function between two points on the x-axis. Unlike indefinite integrals, definite integrals yield a numerical value.

Integral of 1/x²

To find the integral of [Tex]\frac{1}{x^2}[/Tex]​, we can use the power rule for integration.

Function [Tex]\frac{1}{x^2}[/Tex] can be rewritten as x⁻².

So we want to find:

[Tex]\int x^{-2} \, dx[/Tex]

Using the power rule for integration:

[Tex]\int x^n \, dx = \frac{x^{n+1}}{n+1} + C[/Tex]

where n ≠−1. Here, n = −2, so:

[Tex]\int x^{-2} \, dx = \frac{x^{-2 + 1}}{-2 + 1} + C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C[/Tex]

Thus, the integral of [Tex]\frac{1}{x^2}[/Tex]​ is:

[Tex]\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C[/Tex]

where C is the constant of integration.

Conclusion

The integral of [Tex]\frac{1}{x^2}[/Tex]​ can be effectively determined by applying the power rule for integration. By rewriting [Tex]\frac{1}{x^2}​ [/Tex] as [Tex]x^{-2}[/Tex], we simplify the integral to [Tex]\int x^{-2} \, dx[/Tex]. Using the power rule, we find that the antiderivative is -\frac{1}{x}​, plus an integration constant C. Thus, the integral [Tex]\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C[/Tex], where C represents any constant value that could be added to the function, reflecting the general solution to the problem. This result highlights the power rule’s utility in solving integrals involving negative exponents.

Read More:

• Find the Integral of 1/x
• Methods of Integration

Frequently Asked Questions

What is an integral?

An integral is a fundamental concept in calculus that represents the accumulation of quantities, such as areas under curves, total change, and accumulated quantities. There are two main types: indefinite integrals, which find antiderivatives, and definite integrals, which calculate the net accumulation over an interval.

How are integrals used in real life?

Integrals are used in a wide range of applications, including:

• Calculating areas and volumes.
• Determining total accumulated quantities (e.g., distance traveled, mass).
• Solving differential equations in physics and engineering.
• Analyzing growth rates in biology and economics.
• Computing probabilities in statistics.

How are integrals used in physics?

Integrals are used in physics to calculate quantities like work, energy, and electric charge distributions, and to solve differential equations describing physical systems.

What is integration by parts?

A method based on the product rule for differentiation, used to integrate products of functions.

What is partial fraction decomposition?

A technique used to break down rational functions into simpler fractions, making them easier to integrate.