What is integral of 1/x^2? What is the integral of one over x squared?

Understanding the Integral of 1/x²

When we delve into the world of calculus, particularly integration, one of the more straightforward yet essential functions we encounter is ( frac{1}{x^2} ). This function is significant in various practical applications, from physics to economics. Let’s explore the integral of this function and understand it clearly.

The Integral of 1/x²

To compute the integral of ( frac{1}{x^2} ), we can express it in a more convenient form:

$int frac{1}{x^2} , dx = int x^{-2} , dx$

Using the power rule of integration, which states that:

$int x^n , dx = frac{x^{n+1}}{n+1} + C quad text{for} , n neq -1$

we can apply it to our integral where ( n = -2 ):

$int x^{-2} , dx = frac{x^{-2 + 1}}{-2 + 1} + C = frac{x^{-1}}{-1} + C = -frac{1}{x} + C$

Here, ( C ) represents the constant of integration, which is crucial in indefinite integrals as it accounts for all possible antiderivatives.

Conclusion

Thus, the integral of ( frac{1}{x^2} ) is:

$int frac{1}{x^2} , dx = -frac{1}{x} + C$

This result not only gives us the area under the curve ( frac{1}{x^2} ) but also provides insight into the behavior of the function. As ( x ) approaches zero, ( frac{1}{x^2} ) becomes unbounded, which can lead to interesting discussions about limits and improper integrals.

Understanding this integral forms a foundational knowledge that can be applied in more complex integrals and real-world applications, enhancing our problem-solving toolkit in calculus.