Evaluate The Integral: $ \int_0^2 \frac{d X}{\sqrt{8 X-x^2}} $

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Introduction

In this article, we will delve into the world of calculus and evaluate the given integral: $ \int_0^2 \frac{d x}{\sqrt{8 x-x^2}} $. This problem requires a deep understanding of integration techniques, particularly the method of substitution. We will break down the solution into manageable steps, making it easier to follow and understand.

Understanding the Integral

The given integral is $ \int_0^2 \frac{d x}{\sqrt{8 x-x^2}} $. At first glance, this may seem like a daunting task, but with the right approach, we can simplify it and arrive at a solution. The key to solving this integral lies in recognizing the form of the integrand and applying the appropriate substitution.

Recognizing the Form of the Integrand

The integrand can be rewritten as $ \frac{d x}{\sqrt{8 x-x^2}} = \frac{d x}{\sqrt{8 x(1-\frac{x}{8})}} $. This form suggests that we can use the substitution method to simplify the integral.

Applying the Substitution Method

Let's substitute $ u = 8 x(1-\frac{x}{8}) $. This substitution will help us simplify the integral and make it more manageable. To find the derivative of $ u $ with respect to $ x $, we'll use the chain rule.

Finding the Derivative of u

The derivative of $ u $ with respect to $ x $ is given by:

dudx=ddx(8x(1−x8))\frac{d u}{d x} = \frac{d}{d x} (8 x(1-\frac{x}{8}))

Using the product rule and the chain rule, we get:

dudx=8(1−x8)+8x(−18)\frac{d u}{d x} = 8(1-\frac{x}{8}) + 8x(-\frac{1}{8})

Simplifying the expression, we get:

dudx=8−x−x\frac{d u}{d x} = 8 - x - x

dudx=8−2x\frac{d u}{d x} = 8 - 2x

Simplifying the Integral

Now that we have the derivative of $ u $, we can rewrite the integral in terms of $ u $. We'll substitute $ u $ and $ \frac{d u}{d x} $ into the original integral.

∫02dx8x−x2=∫duu\int_0^2 \frac{d x}{\sqrt{8 x-x^2}} = \int \frac{d u}{\sqrt{u}}

Evaluating the Integral

The integral $ \int \frac{d u}{\sqrt{u}} $ is a standard integral that can be evaluated using the power rule of integration.

∫duu=2u+C\int \frac{d u}{\sqrt{u}} = 2 \sqrt{u} + C

Finding the Value of the Integral

Now that we have the antiderivative, we can evaluate the integral by substituting the limits of integration.

∫02dx8x−x2=[2u]02\int_0^2 \frac{d x}{\sqrt{8 x-x^2}} = \left[ 2 \sqrt{u} \right]_0^2

Substituting the limits of integration, we get:

∫02dx8x−x2=22(1−28)−20\int_0^2 \frac{d x}{\sqrt{8 x-x^2}} = 2 \sqrt{2(1-\frac{2}{8})} - 2 \sqrt{0}

Simplifying the expression, we get:

∫02dx8x−x2=212\int_0^2 \frac{d x}{\sqrt{8 x-x^2}} = 2 \sqrt{\frac{1}{2}}

∫02dx8x−x2=2\int_0^2 \frac{d x}{\sqrt{8 x-x^2}} = \sqrt{2}

Conclusion

In this article, we evaluated the given integral $ \int_0^2 \frac{d x}{\sqrt{8 x-x^2}} $ using the substitution method. We recognized the form of the integrand, applied the substitution method, and simplified the integral to arrive at a solution. The final answer is $ \sqrt{2} $.

Final Answer

The final answer is $ \boxed{\sqrt{2}} $.

Additional Tips and Tricks

  • When faced with a difficult integral, try to recognize the form of the integrand and apply the appropriate substitution.
  • Use the chain rule and the product rule to find the derivative of the substitution.
  • Simplify the integral by substituting the limits of integration.
  • Use the power rule of integration to evaluate the integral.

By following these tips and tricks, you'll be able to tackle even the most challenging integrals with confidence.

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Introduction

In our previous article, we evaluated the integral $ \int_0^2 \frac{d x}{\sqrt{8 x-x^2}} $ using the substitution method. In this article, we'll provide a Q&A guide to help you better understand the solution and tackle similar problems.

Q: What is the substitution method?

A: The substitution method is a technique used to simplify integrals by substituting a new variable for a part of the integrand. This method is particularly useful when the integrand can be rewritten in a form that involves a derivative or an integral of a simpler function.

Q: How do I recognize the form of the integrand?

A: To recognize the form of the integrand, look for patterns or structures that suggest a substitution. For example, if the integrand involves a square root or a fraction, try to rewrite it in a form that involves a derivative or an integral of a simpler function.

Q: What is the derivative of u with respect to x?

A: The derivative of $ u $ with respect to $ x $ is given by:

dudx=ddx(8x(1−x8))\frac{d u}{d x} = \frac{d}{d x} (8 x(1-\frac{x}{8}))

Using the product rule and the chain rule, we get:

dudx=8(1−x8)+8x(−18)\frac{d u}{d x} = 8(1-\frac{x}{8}) + 8x(-\frac{1}{8})

Simplifying the expression, we get:

dudx=8−x−x\frac{d u}{d x} = 8 - x - x

dudx=8−2x\frac{d u}{d x} = 8 - 2x

Q: How do I simplify the integral?

A: To simplify the integral, substitute $ u $ and $ \frac{d u}{d x} $ into the original integral. This will help you rewrite the integral in a form that is easier to evaluate.

Q: What is the antiderivative of the integral?

A: The antiderivative of the integral is given by:

∫duu=2u+C\int \frac{d u}{\sqrt{u}} = 2 \sqrt{u} + C

Q: How do I evaluate the integral?

A: To evaluate the integral, substitute the limits of integration into the antiderivative. This will give you the final answer.

Q: What is the final answer?

A: The final answer is $ \boxed{\sqrt{2}} $.

Additional Tips and Tricks

  • When faced with a difficult integral, try to recognize the form of the integrand and apply the appropriate substitution.
  • Use the chain rule and the product rule to find the derivative of the substitution.
  • Simplify the integral by substituting the limits of integration.
  • Use the power rule of integration to evaluate the integral.

By following these tips and tricks, you'll be able to tackle even the most challenging integrals with confidence.

Common Mistakes to Avoid

  • Failing to recognize the form of the integrand and apply the appropriate substitution.
  • Not using the chain rule and the product rule to find the derivative of the substitution.
  • Not simplifying the integral by substituting the limits of integration.
  • Not using the power rule of integration to evaluate the integral.

By avoiding these common mistakes, you'll be able to arrive at the correct solution and build your confidence in tackling challenging integrals.

Conclusion

In this article, we provided a Q&A guide to help you better understand the solution to the integral $ \int_0^2 \frac{d x}{\sqrt{8 x-x^2}} $ and tackle similar problems. By following the tips and tricks outlined in this article, you'll be able to tackle even the most challenging integrals with confidence.