Integrating the Function √x - x/x2: A Comprehensive Guide

In this article, we will explore the process of integrating the function √x - x/x2. We will present and explain multiple methods to achieve this integration, ensuring that our approach aligns with Google's standards for content quality and accessibility. Let's delve into the details.

Integration Techniques for √x - x/x2

The function in question is √x - x/x2, which can be rewritten as 1/sqrt{x} - 1/x. To integrate this, we can use the standard rules of integration for x and 1/x as well as the rule for integrals of the form x^n.

Method 1: Direct Integration

We start with the integral:

int frac{sqrt{x} - x}{x^2}dx int left( frac{1}{xsqrt{x}} - frac{1}{x}right)dx.

This can be split into two separate integrals:

int frac{1}{xsqrt{x}}dx - int frac{1}{x}dx.

The first integral can be simplified to:

int x^{-frac{3}{2}}dx -2x^{-frac{1}{2}} -frac{2}{sqrt{x}}.

The second integral is simply the natural logarithm of the absolute value of x:

int frac{1}{x}dx ln(x).

Combining these results, we get:

int frac{sqrt{x} - x}{x^2}dx -frac{2}{sqrt{x}} - ln(x) C.

Method 2: Simplified Integral Steps

We can also simplify the integral step-by-step as follows:

First, rewrite the integral:

int frac{sqrt{x} - x}{x^2}dx int frac{sqrt{x}}{x^2}dx - int frac{x}{x^2}dx.

For the first integral, use the substitution u x^{1/2}, so du frac{1}{2}x^{-1/2}dx:

int x^{-frac{3}{2}}dx -2x^{-frac{1}{2}} -frac{2}{sqrt{x}}.

For the second integral, use the standard integral of 1/x:

int frac{1}{x}dx ln(x).

Combining both results, we get the final integrated form:

int frac{sqrt{x} - x}{x^2}dx -frac{2}{sqrt{x}} - ln(x) C.

Method 3: Simplifying with Common Fractions

We can simplify the integral by rewriting the function as:

int frac{sqrt{x} - x}{x^2}dx int frac{1}{xsqrt{x}} dx - int frac{1}{x} dx.

Following the steps in Method 1, we get:

int frac{1}{xsqrt{x}} dx -2x^{-frac{1}{2}} -frac{2}{sqrt{x}}.

And:

int frac{1}{x} dx ln(x).

Combining these, we obtain:

int frac{sqrt{x} - x}{x^2}dx -frac{2}{sqrt{x}} - ln(x) C.

Conclusion

In summary, integrating the function √x - x/x2 can be done using various methods, all of which lead to the same final result: -2/√x - ln(x) C. This guide has provided a detailed and step-by-step explanation to ensure clarity and ease of understanding.

References and Further Reading

If you are interested in diving deeper into calculus and integration, check out the following resources:

Advanced Calculus TextbookIntegration Techniques GuideSahus Universe Blog

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