Integrate: $\int X^4 \ln X \, Dx = \square + C$

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Introduction

Integration is a fundamental concept in calculus, and it plays a crucial role in solving problems in various fields, including physics, engineering, and economics. In this article, we will focus on integrating the function $x^4 \ln x$, which is a product of two functions: $x^4$ and $\ln x$. This type of integral is known as a product integral, and it requires the use of integration by parts and other techniques to evaluate.

Background

Before we dive into the integration process, let's review some of the key concepts and formulas that we will need to use. The natural logarithm is a function that is defined as the inverse of the exponential function. It is denoted by $\ln x$ and is defined for all positive real numbers $x$. The derivative of the natural logarithm is given by $\frac{d}{dx} \ln x = \frac{1}{x}$.

Integration by Parts

One of the most powerful techniques for integrating functions is integration by parts. This technique is based on the product rule for differentiation, which states that if $u$ and $v$ are two functions, then the derivative of their product is given by $\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}$. Integration by parts is a way of reversing this process, and it is used to integrate functions that are products of two or more functions.

Applying Integration by Parts

To integrate the function $x^4 \ln x$, we will use integration by parts. We will choose $u = \ln x$ and $dv = x^4 dx$. Then, we have $du = \frac{1}{x} dx$ and $v = \frac{x^5}{5}$. Now, we can apply the formula for integration by parts, which is given by $\int u dv = uv - \int v du$.

Evaluating the Integral

Using the formula for integration by parts, we have:

$$\int x^4 \ln x \, dx = \frac{x^5}{5} \ln x - \int \frac{x^5}{5} \frac{1}{x} \, dx$$

Simplifying the Integral

Now, we can simplify the integral by canceling out the common factor of $x$ in the numerator and denominator:

$$\int x^4 \ln x \, dx = \frac{x^5}{5} \ln x - \frac{1}{5} \int x^4 \, dx$$

Evaluating the Remaining Integral

The remaining integral is a power integral, and it can be evaluated using the formula for the integral of $x^n$, which is given by $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. In this case, we have $n = 4$, so we can evaluate the integral as follows:

$$\int x^4 \, dx = \frac{x^5}{5} + C$$

Combining the Results

Now, we can combine the results of the two integrals to obtain the final answer:

$$\int x^4 \ln x \, dx = \frac{x^5}{5} \ln x - \frac{1}{5} \left( \frac{x^5}{5} + C \right)$$

Simplifying the Final Answer

Finally, we can simplify the final answer by combining like terms:

$$\int x^4 \ln x \, dx = \frac{x^5}{5} \ln x - \frac{x^5}{25} - \frac{C}{5}$$

Conclusion

In this article, we have shown how to integrate the function $x^4 \ln x$ using integration by parts and other techniques. We have also reviewed some of the key concepts and formulas that are used in integration, including the natural logarithm and the derivative. We hope that this article has been helpful in providing a clear and concise explanation of how to integrate this type of function.

Additional Resources

If you are interested in learning more about integration and other topics in calculus, we recommend checking out the following resources:

• Khan Academy: Calculus
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Integration

Final Answer

The final answer is: $\boxed{\frac{x^5}{5} \ln x - \frac{x^5}{25} - \frac{C}{5}}$

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Introduction

In our previous article, we showed how to integrate the function $x^4 \ln x$ using integration by parts and other techniques. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is integration by parts?

A: Integration by parts is a technique used to integrate functions that are products of two or more functions. It is based on the product rule for differentiation, which states that if $u$ and $v$ are two functions, then the derivative of their product is given by $\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}$. Integration by parts is a way of reversing this process.

Q: How do I choose $u$ and $dv$ when using integration by parts?

A: When using integration by parts, you need to choose two functions, $u$ and $dv$, such that $u$ is a function that is easy to integrate, and $dv$ is a function that is easy to differentiate. In the case of the function $x^4 \ln x$, we chose $u = \ln x$ and $dv = x^4 dx$.

Q: What is the formula for integration by parts?

A: The formula for integration by parts is given by $\int u dv = uv - \int v du$. This formula is used to integrate functions that are products of two or more functions.

Q: How do I evaluate the integral $\int x^4 \ln x \, dx$?

A: To evaluate the integral $\int x^4 \ln x \, dx$, we used integration by parts. We chose $u = \ln x$ and $dv = x^4 dx$, and then we applied the formula for integration by parts. The result was $\int x^4 \ln x \, dx = \frac{x^5}{5} \ln x - \frac{1}{5} \int x^4 \, dx$.

Q: How do I evaluate the integral $\int x^4 \, dx$?

A: The integral $\int x^4 \, dx$ is a power integral, and it can be evaluated using the formula for the integral of $x^n$, which is given by $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. In this case, we have $n = 4$, so we can evaluate the integral as follows: $\int x^4 \, dx = \frac{x^5}{5} + C$.

Q: What is the final answer to the integral $\int x^4 \ln x \, dx$?

A: The final answer to the integral $\int x^4 \ln x \, dx$ is $\frac{x^5}{5} \ln x - \frac{x^5}{25} - \frac{C}{5}$.

Q: What are some common mistakes to avoid when using integration by parts?

A: Some common mistakes to avoid when using integration by parts include:

• Choosing $u$ and $dv$ incorrectly
• Failing to apply the formula for integration by parts correctly
• Not simplifying the result correctly

Q: What are some tips for mastering integration by parts?

A: Some tips for mastering integration by parts include:

• Practicing, practicing, practicing
• Paying close attention to the details of the problem
• Using the formula for integration by parts correctly
• Simplifying the result correctly

Conclusion

In this article, we have answered some of the most frequently asked questions about integrating the function $x^4 \ln x$ using integration by parts and other techniques. We hope that this article has been helpful in providing a clear and concise explanation of how to integrate this type of function.

Additional Resources

If you are interested in learning more about integration and other topics in calculus, we recommend checking out the following resources:

• Khan Academy: Calculus
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Integration

Final Answer

The final answer is: $\boxed{\frac{x^5}{5} \ln x - \frac{x^5}{25} - \frac{C}{5}}$