Integration of dx/(1 sqrt(x)) using U-Substitution: A Step-by-Step Guide

Integration is a fundamental concept in calculus, and techniques such as u-substitution play a crucial role in solving complex integral problems. This article focuses on how to find the integral of dx/(1 sqrt(x)) using u-substitution, with detailed step-by-step solutions and explanations.

Problem Statement

The integral in question is:

mathcal{I}(x) int frac{1}{sqrt{x} 1} dx

Step-by-Step Solution

Let's solve this problem using u-substitution. The key is to choose an appropriate substitution, which simplifies the integrand.

Step 1: Substitution

We start by setting:

u sqrt(x)

This implies:

x u^2

And the differential:

dx 2u du

Step 2: Rewrite the Integral

Substituting these into the original integral:

mathcal{I}(x) int frac{1}{sqrt{x} 1} dx int frac{1}{u 1} 2u du

Step 3: Simplify the Integrand

We can further simplify the integrand:

mathcal{I}(x) 2 int frac{u}{u 1} du

Breaking down the integrand, we get:

mathcal{I}(x) 2 int left(1 - frac{1}{u 1} right) du

This can be integrated term by term:

mathcal{I}(x) 2 left( int 1 du - int frac{1}{u 1} du right)

Step 4: Integrate Each Term

Integrating each term:

mathcal{I}(x) 2 left( u - ln|u 1| right) C

Step 5: Back-Substitute

Substitute back ( u sqrt(x) ) and simplify:

mathcal{I}(x) 2 sqrt(x) - 2 ln|sqrt(x) 1| C

Conclusion

In this article, we have demonstrated the detailed process of solving an integral using u-substitution. The key steps involved setting appropriate variables, rewriting the integral, and integrating each term separately. The final solution is:

mathcal{I}(x) 2 sqrt(x) - 2 ln|sqrt(x) 1| C

Additional Notes

U-substitution is a powerful tool in calculus, and it can be applied to various types of integrals. Here are a few more examples to help solidify your understanding:

Example 1: Integrate dx/(1 - sqrt(x))

Let ( u 1 - sqrt(x) ). Then, ( x (1-u)^2 ) and ( dx -2(1-u)du ).

The integral becomes:

int frac{dx}{1 - sqrt(x)} -2 int frac{1-u}{u} du

After integration, you can back-substitute to find the final result.

Example 2: Integrate dx/(sqrt(x) 1)

Let ( u sqrt(x) ). Then, ( x u^2 ) and ( dx 2u du ).

The integral becomes:

int frac{dx}{sqrt(x) 1} 2 int frac{u du}{u 1} 2 int left(1 - frac{1}{u 1} right) du

Integrate each term and back-substitute to get the final answer.

Key Takeaways

Integral: Understanding how to integrate various functions is crucial. u-substitution: This method simplifies integrals by changing variables. Calculus techniques: Mastery in these techniques will make solving integrals easier.

References and Further Reading

For a deeper understanding of u-substitution and other integration techniques, consider reading the following resources:

U-Substitution - The Lamar University website offers detailed explanations and examples. Integration Techniques - MathIsFun provides a comprehensive guide to various integration methods.

By practicing with different integrals and understanding the principles behind u-substitution, you can significantly improve your problem-solving skills in calculus.