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

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Introduction

Mathematical Integration is a fundamental concept in mathematics, and evaluating integrals is a crucial aspect of calculus. In this article, we will focus on evaluating the integral $\int_0^\pi \frac{d x}{\sqrt{8 x - x^2}}$. This integral is a classic example of a definite integral, which is a type of integral that has a specific upper and lower bound. The integral in question is a rational function, which is a function that can be expressed as the ratio of two polynomials.

Understanding the Integral

The integral $\int_0^\pi \frac{d x}{\sqrt{8 x - x^2}}$ can be broken down into several components. The denominator of the integral is a square root function, which is a function that returns the square root of a given value. The numerator of the integral is a constant function, which is a function that returns a constant value. The limits of integration are $0$ and $\pi$, which are the upper and lower bounds of the integral.

Evaluating the Integral

To evaluate the integral, we can use the substitution method, which is a technique used to simplify the integral by substituting a new variable for the original variable. In this case, we can substitute $x = 4 \sin^2 \theta$, which is a trigonometric substitution. This substitution will simplify the integral and make it easier to evaluate.

Trigonometric Substitution

The trigonometric substitution $x = 4 \sin^2 \theta$ is a common technique used to simplify integrals that involve square root functions. This substitution will transform the integral into a form that is easier to evaluate.

Simplifying the Integral

Using the trigonometric substitution $x = 4 \sin^2 \theta$, we can simplify the integral as follows:

$$\int_0^\pi \frac{d x}{\sqrt{8 x - x^2}} = \int_0^{\frac{\pi}{2}} \frac{8 \sin \theta \cos \theta d \theta}{\sqrt{8 (4 \sin^2 \theta) - (4 \sin^2 \theta)^2}}$$

Evaluating the Simplified Integral

The simplified integral can be evaluated using the substitution method. We can substitute $u = \sin \theta$, which will simplify the integral and make it easier to evaluate.

Substitution Method

Using the substitution $u = \sin \theta$, we can simplify the integral as follows:

$$\int_0^{\frac{\pi}{2}} \frac{8 \sin \theta \cos \theta d \theta}{\sqrt{8 (4 \sin^2 \theta) - (4 \sin^2 \theta)^2}} = \int_0^1 \frac{8 u du}{\sqrt{8 (4 u^2) - (4 u^2)^2}}$$

Evaluating the Final Integral

The final integral can be evaluated using the power rule of integration, which is a technique used to integrate functions that involve powers. In this case, we can use the power rule to integrate the function and evaluate the integral.

Power Rule of Integration

Using the power rule of integration, we can evaluate the final integral as follows:

$$\int_0^1 \frac{8 u du}{\sqrt{8 (4 u^2) - (4 u^2)^2}} = \left[ \frac{8}{3} \sqrt{8 (4 u^2) - (4 u^2)^2} \right]_0^1$$

Evaluating the Result

The final result can be evaluated by substituting the upper and lower bounds of the integral. In this case, we can substitute $u = 1$ and $u = 0$ to evaluate the result.

Final Result

The final result is:

$$\left[ \frac{8}{3} \sqrt{8 (4 u^2) - (4 u^2)^2} \right]_0^1 = \frac{8}{3} \left( \sqrt{8 (4 (1)^2) - (4 (1)^2)^2} - \sqrt{8 (4 (0)^2) - (4 (0)^2)^2} \right)$$

Simplifying the Final Result

The final result can be simplified by evaluating the square roots and simplifying the expression.

Simplified Final Result

The simplified final result is:

$$\frac{8}{3} \left( \sqrt{8 (4 (1)^2) - (4 (1)^2)^2} - \sqrt{8 (4 (0)^2) - (4 (0)^2)^2} \right) = \frac{8}{3} \left( \sqrt{32 - 16} - \sqrt{0} \right)$$

Final Answer

The final answer is:

$$\frac{8}{3} \left( \sqrt{16} - \sqrt{0} \right) = \frac{8}{3} \left( 4 - 0 \right) = \frac{32}{3}$$

Conclusion

In this article, we evaluated the integral $\int_0^\pi \frac{d x}{\sqrt{8 x - x^2}}$ using the substitution method and the power rule of integration. The final result is $\frac{32}{3}$, which is the value of the integral. This example demonstrates the importance of using trigonometric substitutions and the power rule of integration to simplify and evaluate integrals.

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Introduction

In our previous article, we evaluated the integral $\int_0^\pi \frac{d x}{\sqrt{8 x - x^2}}$ using the substitution method and the power rule of integration. In this article, we will answer some common questions related to this integral and provide additional insights into the evaluation process.

Q: What is the significance of the limits of integration in this problem?

A: The limits of integration, $0$ and $\pi$, are crucial in this problem. They define the interval over which the integral is evaluated, and they also affect the substitution used to simplify the integral.

Q: Why was the trigonometric substitution $x = 4 \sin^2 \theta$ used in this problem?

A: The trigonometric substitution $x = 4 \sin^2 \theta$ was used to simplify the integral and make it easier to evaluate. This substitution transformed the integral into a form that is more amenable to integration.

Q: How does the power rule of integration apply to this problem?

A: The power rule of integration is used to integrate functions that involve powers. In this problem, the power rule is used to integrate the function $\frac{8 u du}{\sqrt{8 (4 u^2) - (4 u^2)^2}}$.

Q: What is the final result of the integral?

A: The final result of the integral is $\frac{32}{3}$.

Q: Can this integral be evaluated using other methods?

A: Yes, this integral can be evaluated using other methods, such as the method of partial fractions or the method of substitution with a different substitution.

Q: What are some common mistakes to avoid when evaluating this integral?

A: Some common mistakes to avoid when evaluating this integral include:

• Failing to simplify the integral using the substitution method
• Failing to apply the power rule of integration correctly
• Failing to evaluate the integral over the correct interval

Q: How can this integral be applied in real-world problems?

A: This integral can be applied in real-world problems involving optimization, physics, and engineering. For example, it can be used to model the motion of an object under the influence of a force.

Q: What are some additional resources for learning more about this integral?

A: Some additional resources for learning more about this integral include:

• Calculus textbooks, such as "Calculus" by Michael Spivak
• Online resources, such as Khan Academy and MIT OpenCourseWare
• Research papers and articles on the topic of integration and optimization

Conclusion

In this article, we answered some common questions related to the integral $\int_0^\pi \frac{d x}{\sqrt{8 x - x^2}}$ and provided additional insights into the evaluation process. We also discussed some common mistakes to avoid and provided some additional resources for learning more about this integral.