Use A Substitution To Express The Integrand As A Rational Function And Then Evaluate The Integral:$\int_{36}^{64} \frac{\sqrt{x}}{x-4} \, Dx$

Introduction

In this article, we will explore the process of using a substitution to express the integrand as a rational function and then evaluate the integral. This technique is a powerful tool in calculus, allowing us to simplify complex integrals and make them more manageable.

The Problem

The given problem is to evaluate the integral:

$$\int_{36}^{64} \frac{\sqrt{x}}{x-4} \, dx$$

This integral appears to be a complex one, with a square root in the numerator and a linear term in the denominator. We will use a substitution to simplify this integral and make it more tractable.

Step 1: Identify a Suitable Substitution

To simplify the integral, we need to identify a suitable substitution that will allow us to express the integrand as a rational function. In this case, we can try substituting $u = x - 4$. This will simplify the denominator of the integrand and make it easier to work with.

Step 2: Compute the Derivative of the Substitution

To compute the derivative of the substitution, we need to differentiate $u = x - 4$ with respect to $x$. This gives us:

$$\frac{du}{dx} = 1$$

Step 3: Express the Integrand in Terms of the Substitution

Now that we have the derivative of the substitution, we can express the integrand in terms of $u$. We have:

$$\frac{\sqrt{x}}{x-4} = \frac{\sqrt{x+4}}{u}$$

Step 4: Simplify the Integrand

We can simplify the integrand further by multiplying the numerator and denominator by $\sqrt{x+4}$. This gives us:

$$\frac{\sqrt{x+4}}{u} = \frac{u+4}{u^2}$$

Step 5: Evaluate the Integral

Now that we have simplified the integrand, we can evaluate the integral. We have:

$$\int_{36}^{64} \frac{\sqrt{x}}{x-4} \, dx = \int_{32}^{60} \frac{u+4}{u^2} \, du$$

To evaluate this integral, we can use the power rule of integration, which states that:

$$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$$

Applying this rule to our integral, we get:

$$\int_{32}^{60} \frac{u+4}{u^2} \, du = \int_{32}^{60} \frac{1}{u} + \frac{4}{u^2} \, du$$

Evaluating this integral, we get:

$$\int_{32}^{60} \frac{1}{u} + \frac{4}{u^2} \, du = \ln|u| - \frac{4}{u} \Big|_{32}^{60}$$

Simplifying this expression, we get:

$$\ln|u| - \frac{4}{u} \Big|_{32}^{60} = \ln|60| - \frac{4}{60} - \ln|32| + \frac{4}{32}$$

Combining like terms, we get:

$$\ln|60| - \frac{4}{60} - \ln|32| + \frac{4}{32} = \ln\left(\frac{60}{32}\right) - \frac{1}{15}$$

Simplifying this expression, we get:

$$\ln\left(\frac{60}{32}\right) - \frac{1}{15} = \ln\left(\frac{15}{8}\right) - \frac{1}{15}$$

Conclusion

In this article, we used a substitution to express the integrand as a rational function and then evaluated the integral. We identified a suitable substitution, computed the derivative of the substitution, expressed the integrand in terms of the substitution, simplified the integrand, and finally evaluated the integral. The final answer is $\boxed{\ln\left(\frac{15}{8}\right) - \frac{1}{15}}$.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Note

Introduction

In our previous article, we explored the process of using a substitution to express the integrand as a rational function and then evaluate the integral. In this article, we will answer some common questions related to this topic.

Q: What is a substitution in calculus?

A: A substitution in calculus is a technique used to simplify a complex integral by replacing the variable of integration with a new variable. This new variable is often a function of the original variable.

Q: Why do we need to use a substitution?

A: We need to use a substitution to simplify complex integrals that cannot be evaluated directly. By replacing the variable of integration with a new variable, we can make the integral more manageable and easier to evaluate.

Q: What are some common substitutions used in calculus?

A: Some common substitutions used in calculus include:

• $u = x - a$
• $u = x + a$
• $u = ax$
• $u = \frac{1}{x}$

Q: How do we choose a suitable substitution?

A: To choose a suitable substitution, we need to identify a function of the original variable that will simplify the integral. This function should be a simple one that can be easily integrated.

Q: What are some common mistakes to avoid when using a substitution?

A: Some common mistakes to avoid when using a substitution include:

• Not checking if the substitution is valid
• Not computing the derivative of the substitution
• Not expressing the integrand in terms of the substitution
• Not simplifying the integrand

Q: Can we use a substitution to evaluate any type of integral?

A: No, we cannot use a substitution to evaluate any type of integral. Substitutions are most effective for integrals that have a complex integrand or a complex variable of integration.

Q: Are there any other techniques for evaluating integrals besides substitution?

A: Yes, there are other techniques for evaluating integrals besides substitution. Some common techniques include:

• Integration by parts
• Integration by partial fractions
• Integration by trigonometric substitution
• Integration by hyperbolic substitution

Q: How do we know when to use a substitution and when to use another technique?

A: To determine when to use a substitution and when to use another technique, we need to analyze the integral and identify the most effective method for evaluating it. This may involve trying out different techniques and seeing which one works best.

Conclusion

In this article, we answered some common questions related to using a substitution to express the integrand as a rational function and then evaluate the integral. We discussed the importance of choosing a suitable substitution, computing the derivative of the substitution, expressing the integrand in terms of the substitution, and simplifying the integrand. We also discussed some common mistakes to avoid and other techniques for evaluating integrals.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Note

The questions and answers in this article are based on common topics and questions related to using a substitution to express the integrand as a rational function and then evaluate the integral. The answers are provided in a clear and concise manner, with explanations and examples to help illustrate the concepts.