Evaluate The Integral:$\[ \int \frac{x \, D X}{\sqrt{3x^2 + 5}} \\]

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Introduction

Mathematical Integration is a fundamental concept in mathematics, and it plays a crucial role in various fields, including physics, engineering, and economics. In this article, we will focus on evaluating the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$. This integral is a classic example of a trigonometric substitution, which is a technique used to simplify and evaluate integrals.

Background

The integral in question is a rational function of the form $\frac{x}{\sqrt{3x^2 + 5}}$. To evaluate this integral, we need to use a substitution method. The substitution method is a technique used to simplify and evaluate integrals by replacing the variable of integration with a new variable.

Substitution Method

The substitution method involves replacing the variable of integration with a new variable that is a function of the original variable. In this case, we can use the substitution $u = 3x^2 + 5$. This substitution will simplify the integral and make it easier to evaluate.

Step 1: Find the Derivative of u

To use the substitution method, we need to find the derivative of u with respect to x. The derivative of u is given by:

$$\frac{d u}{d x} = 6x$$

Step 2: Substitute u into the Integral

Now that we have found the derivative of u, we can substitute u into the integral. We get:

$$\int \frac{x \, d x}{\sqrt{3x^2 + 5}} = \int \frac{1}{2\sqrt{u}} \, d u$$

Step 3: Evaluate the Integral

Now that we have substituted u into the integral, we can evaluate it. We get:

$$\int \frac{1}{2\sqrt{u}} \, d u = \frac{1}{2} \int u^{-\frac{1}{2}} \, d u$$

Step 4: Use the Power Rule of Integration

The power rule of integration states that:

$$\int x^n \, d x = \frac{x^{n+1}}{n+1} + C$$

We can use this rule to evaluate the integral:

$$\frac{1}{2} \int u^{-\frac{1}{2}} \, d u = \frac{1}{2} \cdot \frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C$$

Step 5: Simplify the Result

Now that we have evaluated the integral, we can simplify the result. We get:

$$\frac{1}{2} \cdot \frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C = \sqrt{u} + C$$

Step 6: Substitute Back u = 3x^2 + 5

Now that we have simplified the result, we can substitute back u = 3x^2 + 5. We get:

$$\sqrt{u} + C = \sqrt{3x^2 + 5} + C$$

Conclusion

In this article, we have evaluated the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$ using the substitution method. We have shown that the integral can be simplified and evaluated using the substitution $u = 3x^2 + 5$. The final result is $\sqrt{3x^2 + 5} + C$, where C is the constant of integration.

Final Answer

The final answer is $\boxed{\sqrt{3x^2 + 5} + C}$.

Related Topics

• Trigonometric Substitution: This is a technique used to simplify and evaluate integrals by replacing the variable of integration with a new variable.
• Power Rule of Integration: This is a rule used to evaluate integrals of the form $\int x^n \, d x$.
• Constant of Integration: This is a constant that is added to the result of an integral to make it a general solution.

References

• Calculus: This is a branch of mathematics that deals with the study of rates of change and accumulation.
• Integration: This is a process of finding the antiderivative of a function.
• Substitution Method: This is a technique used to simplify and evaluate integrals by replacing the variable of integration with a new variable.

Future Work

• Evaluating Integrals: This is a topic that is closely related to the evaluation of the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$.
• Trigonometric Substitution: This is a technique that can be used to simplify and evaluate integrals.
• Power Rule of Integration: This is a rule that can be used to evaluate integrals of the form $\int x^n \, d x$.

Limitations

• This article assumes that the reader has a basic understanding of calculus and integration.
• The substitution method is a technique that can be used to simplify and evaluate integrals, but it may not always be the most efficient method.
• The power rule of integration is a rule that can be used to evaluate integrals of the form $\int x^n \, d x$, but it may not always be the most efficient method.
  

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Introduction

In our previous article, we evaluated the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$ using the substitution method. In this article, we will answer some frequently asked questions about this integral.

Q: What is the substitution method?

A: The substitution method is a technique used to simplify and evaluate integrals by replacing the variable of integration with a new variable. In the case of the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$, we used the substitution $u = 3x^2 + 5$.

Q: Why did we use the substitution $u = 3x^2 + 5$?

A: We used the substitution $u = 3x^2 + 5$ because it simplified the integral and made it easier to evaluate. The derivative of u with respect to x is $6x$, which is a simple expression.

Q: What is the power rule of integration?

A: The power rule of integration is a rule used to evaluate integrals of the form $\int x^n \, d x$. The power rule states that:

$$\int x^n \, d x = \frac{x^{n+1}}{n+1} + C$$

Q: How did we use the power rule of integration to evaluate the integral?

A: We used the power rule of integration to evaluate the integral $\int \frac{1}{2\sqrt{u}} \, d u$. We first substituted u into the integral, and then we used the power rule to evaluate the integral.

Q: What is the constant of integration?

A: The constant of integration is a constant that is added to the result of an integral to make it a general solution. In the case of the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$, the constant of integration is C.

Q: Why is the constant of integration necessary?

A: The constant of integration is necessary because it allows us to find the general solution of the integral. Without the constant of integration, we would only be able to find the particular solution of the integral.

Q: Can we use other methods to evaluate the integral?

A: Yes, we can use other methods to evaluate the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$. For example, we can use the trigonometric substitution method or the hyperbolic substitution method.

Q: What are the advantages and disadvantages of using the substitution method?

A: The advantages of using the substitution method are that it can simplify the integral and make it easier to evaluate. The disadvantages of using the substitution method are that it may not always be the most efficient method, and it may require a lot of work to find the correct substitution.

Q: What are the advantages and disadvantages of using the power rule of integration?

A: The advantages of using the power rule of integration are that it can simplify the integral and make it easier to evaluate. The disadvantages of using the power rule of integration are that it may not always be the most efficient method, and it may require a lot of work to find the correct power.

Q: Can we use the substitution method and the power rule of integration together?

A: Yes, we can use the substitution method and the power rule of integration together to evaluate the integral. This can be a powerful technique for simplifying and evaluating integrals.

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

A: Some common mistakes to avoid when using the substitution method are:

• Not checking if the substitution is valid
• Not finding the correct derivative of the substitution
• Not simplifying the integral correctly
• Not using the correct power rule of integration

Q: What are some common mistakes to avoid when using the power rule of integration?

A: Some common mistakes to avoid when using the power rule of integration are:

• Not checking if the power rule is valid
• Not finding the correct derivative of the power
• Not simplifying the integral correctly
• Not using the correct substitution method

Conclusion

In this article, we have answered some frequently asked questions about the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$. We have discussed the substitution method, the power rule of integration, and the constant of integration. We have also discussed some common mistakes to avoid when using these techniques.

Final Answer

The final answer is $\boxed{\sqrt{3x^2 + 5} + C}$.

Related Topics

• Trigonometric Substitution: This is a technique used to simplify and evaluate integrals by replacing the variable of integration with a new variable.
• Power Rule of Integration: This is a rule used to evaluate integrals of the form $\int x^n \, d x$.
• Constant of Integration: This is a constant that is added to the result of an integral to make it a general solution.

References

• Calculus: This is a branch of mathematics that deals with the study of rates of change and accumulation.
• Integration: This is a process of finding the antiderivative of a function.
• Substitution Method: This is a technique used to simplify and evaluate integrals by replacing the variable of integration with a new variable.

Future Work

• Evaluating Integrals: This is a topic that is closely related to the evaluation of the integral $\int \frac{x \, d x}{\sqrt{3x^2 + 5}}$.
• Trigonometric Substitution: This is a technique that can be used to simplify and evaluate integrals.
• Power Rule of Integration: This is a rule that can be used to evaluate integrals of the form $\int x^n \, d x$.

Limitations

• This article assumes that the reader has a basic understanding of calculus and integration.
• The substitution method is a technique that can be used to simplify and evaluate integrals, but it may not always be the most efficient method.
• The power rule of integration is a rule that can be used to evaluate integrals of the form $\int x^n \, d x$, but it may not always be the most efficient method.