Evaluating the Integral of sqrt{x^21} with Change of Variables

In this article, we will explore the process of evaluating the integral of sqrt{x^21} using a change of variables technique. This method is a powerful tool in calculus and is often used to simplify integrands, making them easier to solve. We will follow a step-by-step approach to understand the process and apply it practically.

Introduction

The integral we want to evaluate is:

2int_{0}^{1} sqrt{x^{4}x^{2}} dx  int_{0}^{1} 2x sqrt{x^{21}} dx

At first glance, this may seem like a challenging problem, but by using a change of variables technique, we can simplify it and find the solution.

Step-by-Step Solution

Step 1: Simplify the Integral

First, we can simplify the integral by combining the exponents:

2int_{0}^{1} sqrt{x^4x^2} dx  int_{0}^{1} sqrt{x^{6}} dx  int_{0}^{1} x^{3} dx

This simplification shows that we are now focusing on evaluating the integral of a polynomial function.

Step 2: Change of Variables

Next, we introduce a change of variables to simplify the integral further. Let:

u  x^{21}

This implies:

du  21x^{20} dx

However, we need a simpler substitution. We can use:

u  x^2

This substitution will simplify our integral:

du  2x dx

Now we can rewrite the integral in terms of u:

int_{0}^{1} 2x sqrt{x^{21}} dx  int_{0}^{1} x^{21} dx  int_{1}^{2} u^{frac{1}{2}} cdot frac{du}{2x}

Since (u x^2), we have (x sqrt{u}), and thus (dx frac{du}{2sqrt{u}}). The integral becomes:

int_{1}^{2} u^{frac{1}{2}} cdot frac{du}{2sqrt{u}}  frac{1}{2} int_{1}^{2} u^{frac{1}{2}} cdot frac{du}{sqrt{u}}  frac{1}{2} int_{1}^{2} u^{frac{1}{2} - frac{1}{2}} du  frac{1}{2} int_{1}^{2} du

Step 3: Evaluate the Integral

The integral simplifies to:

frac{1}{2} int_{1}^{2} du  frac{1}{2} [u]_{1}^{2}  frac{1}{2} [2 - 1]  frac{1}{2}

This is not the complete solution. Let's revisit and correctly solve the integral using the correct substitution:

u  x^{21}

Then:

du  21x^{20} dx

And the integral transforms to:

frac{2}{21} int u^{frac{1}{2}} du  frac{2}{21} cdot frac{2}{3} u^{frac{3}{2}}  left[ frac{4}{63} u^{frac{3}{2}} right]_{1}^{2}

Final Calculation

Substituting the limits, we get:

frac{4}{63} (2^{frac{3}{2}} - 1^{frac{3}{2}})  frac{4}{63} (2sqrt{2} - 1)

This gives us the final answer:

frac{4}{63} (2sqrt{2} - 1)

Conclusion

In conclusion, the integral of sqrt{x^21} evaluated using a change of variables technique is:

int_{0}^{1} 2x sqrt{x^{21}} dx  frac{4}{63} (2sqrt{2} - 1)

This method of change of variables is a powerful tool in solving integrals and can be applied to a wide range of problems. Understanding and practicing such techniques is crucial for students and professionals in mathematics and related fields.

Resources

For more practice and detailed explanations, refer to the following resources:

Mathematics Department Resources Lamar University Calculus II