Integration of 2x/1 sqrt(x^2 1): A Step-by-Step Guide with Multiple Substitutions

Integration is a fundamental concept in calculus that can sometimes lead to complex expressions. In this article, we will explore the integration of the expression 2x/(1 sqrt(x^2 1)) using multiple substitution methods. This process can be broken down into several steps, making it easier to manage and understand.

1. Introduction

Understanding the steps involved in solving the integral of 2x/(1 sqrt(x^2 1)) can greatly enhance your calculus skills. This guide will walk you through each step, from initial substitution to simplification and eventual solution. We will also explore alternative methods that use multiple substitutions to simplify the problem.

2. The First Substitution

Our initial step is to make the substitution u x^2 1. This substitution can simplify the square root term in the denominator.

u x^2 1

Then du/dx 2x and du 2xdx.

Now, we are left to integrate du/1 sqrt{u}.

3. The Second Substitution

The next step is to use another substitution to further simplify the integral. We let v sqrt{u}.

u v^2

du 2vdv.

We are now left with the integral 2vdv/1 v.

4. The Third Substitution

The final step involves one more substitution. Let w 1-v.

dw/dv 1 and dw dv.

In terms of w, the integral becomes 2(w-1)/w dw or 2(1 - 1/w) dw.

The integral is now much easier to solve.

5. Solving the Integral

By integrating 2(1 - 1/w) dw, we get:

2w - 2ln(w) C_1, where C_1 is a constant of integration.

Substituting back w 1-v 1-sqrt{u} 1-sqrt{1 x^2}, we obtain:

2(1-sqrt{1 x^2}) - 2ln(1-sqrt{1 x^2}) C_1.

Finally, substituting back v sqrt{u} sqrt{1 x^2}, we get:

2(1-sqrt{1 x^2}) - 2ln(1-sqrt{1 x^2}) C_2, where C_2 C_1 2.

This simplifies to:

2(1-sqrt{1 x^2}) - 2ln(1-sqrt{1 x^2}) C, where C is the final constant of integration.

6. Simplification and Final Result

After all the substitutions and integrations, the final result for the integral of 2x/(1 sqrt{x^2 1}) dx is:

2sqrt{x^2 1} - 2ln(1 sqrt{x^2 1}) C

7. Conclusion

In conclusion, the integral of 2x/(1 sqrt{x^2 1}) is:

2sqrt{x^2 1} - 2ln(1 sqrt{x^2 1}) C

This method demonstrates the power of multiple substitutions in solving complex integrals. Understanding these techniques can significantly improve your problem-solving skills in calculus.