Evaluate The Integral $\int 4x \sin 5x \, Dx$.Choose The Correct Method Below:A. Substitution With $u=-\frac{1}{5} \cos (5x$\]B. Integration By PartsC. Partial FractionsD. Substitution With $u= \sin (5x$\]

=====================================================

Introduction

---

When it comes to evaluating integrals, there are several methods to choose from, each with its own strengths and weaknesses. In this article, we will explore the correct method for evaluating the integral $\int 4x \sin 5x \, dx$. We will examine four different options: substitution with $u=-\frac{1}{5} \cos (5x)$, integration by parts, partial fractions, and substitution with $u= \sin (5x)$. By the end of this article, you will have a clear understanding of which method to use and why.

Option A: Substitution with $u=-\frac{1}{5} \cos (5x)$

---

The Method

Substitution is a powerful method for evaluating integrals. It involves replacing a part of the integral with a new variable, which can simplify the integral and make it easier to evaluate. In this case, we can substitute $u=-\frac{1}{5} \cos (5x)$, which will allow us to rewrite the integral in terms of $u$.

The Steps

1. Let $u=-\frac{1}{5} \cos (5x)$.
2. Find $\frac{du}{dx}$, which is equal to $-\frac{5}{5} \sin (5x) = -\sin (5x)$.
3. Rewrite the integral in terms of $u$: $\int 4x \sin 5x \, dx = \int -4x \frac{du}{dx} \, dx$.
4. Evaluate the integral: $\int -4x \frac{du}{dx} \, dx = -4 \int x \, du$.
5. Use the power rule of integration to evaluate the integral: $-4 \int x \, du = -4 \cdot \frac{x^2}{2} + C = -2x^2 + C$.

The Result

The result of using substitution with $u=-\frac{1}{5} \cos (5x)$ is $-2x^2 + C$.

Option B: Integration by Parts

---

The Method

Integration by parts is a method for evaluating integrals that involves differentiating one function and integrating the other. It is a powerful method for evaluating integrals that involve products of functions.

The Steps

1. Choose two functions, $u$ and $v$, to use for integration by parts.
2. Differentiate $u$ and integrate $v$: $\int u \, dv = uv - \int v \, du$.
3. Evaluate the integral: $\int 4x \sin 5x \, dx = \int 4x \sin 5x \, dx = 4x \int \sin 5x \, dx - \int 4 \int \sin 5x \, dx \, dx$.
4. Use the power rule of integration to evaluate the integral: $4x \int \sin 5x \, dx - \int 4 \int \sin 5x \, dx \, dx = -\frac{4x}{5} \cos 5x + \frac{4}{25} \sin 5x + C$.

The Result

The result of using integration by parts is $-\frac{4x}{5} \cos 5x + \frac{4}{25} \sin 5x + C$.

Option C: Partial Fractions

---

The Method

Partial fractions is a method for evaluating integrals that involves breaking down a rational function into simpler fractions.

The Steps

1. Break down the rational function into simpler fractions: $\frac{4x}{\sin 5x} = \frac{A}{\sin 5x} + \frac{B}{\cos 5x}$.
2. Find the values of $A$ and $B$ by equating the numerator of the original function with the numerator of the partial fractions.
3. Evaluate the integral: $\int \frac{4x}{\sin 5x} \, dx = \int \frac{A}{\sin 5x} \, dx + \int \frac{B}{\cos 5x} \, dx$.
4. Use the power rule of integration to evaluate the integral: $\int \frac{A}{\sin 5x} \, dx + \int \frac{B}{\cos 5x} \, dx = -\frac{A}{5} \cos 5x + \frac{B}{5} \sin 5x + C$.

The Result

The result of using partial fractions is $-\frac{A}{5} \cos 5x + \frac{B}{5} \sin 5x + C$.

Option D: Substitution with $u= \sin (5x)$

---

The Method

Substitution is a powerful method for evaluating integrals. It involves replacing a part of the integral with a new variable, which can simplify the integral and make it easier to evaluate. In this case, we can substitute $u= \sin (5x)$, which will allow us to rewrite the integral in terms of $u$.

The Steps

1. Let $u= \sin (5x)$.
2. Find $\frac{du}{dx}$, which is equal to $5 \cos (5x)$.
3. Rewrite the integral in terms of $u$: $\int 4x \sin 5x \, dx = \int 4x \frac{du}{dx} \, dx$.
4. Evaluate the integral: $\int 4x \frac{du}{dx} \, dx = 4 \int x \, du$.
5. Use the power rule of integration to evaluate the integral: $4 \int x \, du = 4 \cdot \frac{x^2}{2} + C = 2x^2 + C$.

The Result

The result of using substitution with $u= \sin (5x)$ is $2x^2 + C$.

Conclusion

---

In this article, we have evaluated the integral $\int 4x \sin 5x \, dx$ using four different methods: substitution with $u=-\frac{1}{5} \cos (5x)$, integration by parts, partial fractions, and substitution with $u= \sin (5x)$. We have seen that each method has its own strengths and weaknesses, and that the correct method to use depends on the specific integral and the skills of the person evaluating it. By the end of this article, you should have a clear understanding of which method to use and why.

Final Answer

---

The final answer is $\boxed{-2x^2 + C}$.

Note: The final answer is the result of using substitution with $u=-\frac{1}{5} \cos (5x)$, which is the correct method for evaluating the integral $\int 4x \sin 5x \, dx$.

$$

=====================================================

Introduction

---

In our previous article, we evaluated the integral $\int 4x \sin 5x \, dx$ using four different methods: substitution with $u=-\frac{1}{5} \cos (5x)$, integration by parts, partial fractions, and substitution with $u= \sin (5x)$. In this article, we will answer some common questions that readers may have about evaluating this integral.

Q: What is the correct method for evaluating the integral $\int 4x \sin 5x \, dx$?

---

A: The correct method for evaluating the integral $\int 4x \sin 5x \, dx$ is substitution with $u=-\frac{1}{5} \cos (5x)$. This method is the most straightforward and efficient way to evaluate this integral.

Q: Why is substitution with $u=-\frac{1}{5} \cos (5x)$ the correct method?

---

A: Substitution with $u=-\frac{1}{5} \cos (5x)$ is the correct method because it allows us to rewrite the integral in terms of $u$, which simplifies the integral and makes it easier to evaluate. This method is also the most efficient way to evaluate this integral, as it requires the fewest number of steps.

Q: What are the steps for evaluating the integral $\int 4x \sin 5x \, dx$ using substitution with $u=-\frac{1}{5} \cos (5x)$?

---

A: The steps for evaluating the integral $\int 4x \sin 5x \, dx$ using substitution with $u=-\frac{1}{5} \cos (5x)$ are:

1. Let $u=-\frac{1}{5} \cos (5x)$.
2. Find $\frac{du}{dx}$, which is equal to $-\frac{5}{5} \sin (5x) = -\sin (5x)$.
3. Rewrite the integral in terms of $u$: $\int 4x \sin 5x \, dx = \int -4x \frac{du}{dx} \, dx$.
4. Evaluate the integral: $\int -4x \frac{du}{dx} \, dx = -4 \int x \, du$.
5. Use the power rule of integration to evaluate the integral: $-4 \int x \, du = -4 \cdot \frac{x^2}{2} + C = -2x^2 + C$.

Q: What are the advantages and disadvantages of using integration by parts to evaluate the integral $\int 4x \sin 5x \, dx$?

---

A: The advantages of using integration by parts to evaluate the integral $\int 4x \sin 5x \, dx$ are:

• It allows us to evaluate the integral in terms of the product of two functions.
• It is a powerful method for evaluating integrals that involve products of functions.

The disadvantages of using integration by parts to evaluate the integral $\int 4x \sin 5x \, dx$ are:

• It requires us to differentiate one function and integrate the other, which can be time-consuming and difficult.
• It is not the most efficient method for evaluating this integral.

Q: What are the advantages and disadvantages of using partial fractions to evaluate the integral $\int 4x \sin 5x \, dx$?

---

A: The advantages of using partial fractions to evaluate the integral $\int 4x \sin 5x \, dx$ are:

• It allows us to break down the rational function into simpler fractions.
• It is a powerful method for evaluating integrals that involve rational functions.

The disadvantages of using partial fractions to evaluate the integral $\int 4x \sin 5x \, dx$ are:

• It requires us to find the values of $A$ and $B$ by equating the numerator of the original function with the numerator of the partial fractions, which can be time-consuming and difficult.
• It is not the most efficient method for evaluating this integral.

Q: What are the advantages and disadvantages of using substitution with $u= \sin (5x)$ to evaluate the integral $\int 4x \sin 5x \, dx$?

---

A: The advantages of using substitution with $u= \sin (5x)$ to evaluate the integral $\int 4x \sin 5x \, dx$ are:

• It allows us to rewrite the integral in terms of $u$, which simplifies the integral and makes it easier to evaluate.
• It is a powerful method for evaluating integrals that involve trigonometric functions.

The disadvantages of using substitution with $u= \sin (5x)$ to evaluate the integral $\int 4x \sin 5x \, dx$ are:

• It requires us to find $\frac{du}{dx}$, which can be time-consuming and difficult.
• It is not the most efficient method for evaluating this integral.

Conclusion

---

In this article, we have answered some common questions that readers may have about evaluating the integral $\int 4x \sin 5x \, dx$. We have discussed the correct method for evaluating this integral, as well as the advantages and disadvantages of using different methods. By the end of this article, you should have a clear understanding of how to evaluate this integral and which method to use.

Final Answer

---

The final answer is $\boxed{-2x^2 + C}$.

Note: The final answer is the result of using substitution with $u=-\frac{1}{5} \cos (5x)$, which is the correct method for evaluating the integral $\int 4x \sin 5x \, dx$.