Evaluate The Integral:$\[ \int \left(x^3 - 3x\right) \, Dx \\]

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Introduction

In this article, we will delve into the world of calculus and evaluate the given integral. The integral in question is $\int \left(x^3 - 3x\right) \, dx$. This is a fundamental problem in calculus, and understanding how to solve it is crucial for further studies in mathematics and physics. We will break down the solution step by step, using various techniques to simplify the integral and arrive at the final answer.

Understanding the Integral

Before we begin, let's understand what the integral represents. The integral of a function $f(x)$ with respect to $x$ is denoted as $\int f(x) \, dx$. It represents the area under the curve of $f(x)$ between two points $a$ and $b$. In this case, we are asked to find the integral of the function $x^3 - 3x$ with respect to $x$.

Step 1: Break Down the Integral

To evaluate the integral, we can break it down into two separate integrals: $\int x^3 \, dx$ and $\int -3x \, dx$. This is because the integral of a sum is equal to the sum of the integrals.

Step 2: Evaluate the First Integral

The first integral is $\int x^3 \, dx$. To evaluate this, we can use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. In this case, $n = 3$, so we have:

$$\int x^3 \, dx = \frac{x^4}{4} + C$$

Step 3: Evaluate the Second Integral

The second integral is $\int -3x \, dx$. To evaluate this, we can use the constant multiple rule of integration, which states that $\int af(x) \, dx = a\int f(x) \, dx$. In this case, $a = -3$, so we have:

$$\int -3x \, dx = -3\int x \, dx$$

Step 4: Evaluate the Integral of $x$

To evaluate the integral of $x$, we can use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. In this case, $n = 1$, so we have:

$$\int x \, dx = \frac{x^2}{2} + C$$

Step 5: Combine the Results

Now that we have evaluated both integrals, we can combine the results to get the final answer:

$$\int \left(x^3 - 3x\right) \, dx = \frac{x^4}{4} - 3\left(\frac{x^2}{2}\right) + C$$

Simplifying the Result

We can simplify the result by combining like terms:

$$\int \left(x^3 - 3x\right) \, dx = \frac{x^4}{4} - \frac{3x^2}{2} + C$$

Conclusion

In this article, we evaluated the integral $\int \left(x^3 - 3x\right) \, dx$. We broke down the integral into two separate integrals, evaluated each one using the power rule and constant multiple rule of integration, and combined the results to get the final answer. The final answer is $\frac{x^4}{4} - \frac{3x^2}{2} + C$, where $C$ is the constant of integration.

Final Answer

The final answer is $\boxed{\frac{x^4}{4} - \frac{3x^2}{2} + C}$.

Related Topics

• Power rule of integration
• Constant multiple rule of integration
• Integration by substitution
• Integration by parts

References

• [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
• [2] Anton, H. (2017). Calculus: Early Transcendentals. John Wiley & Sons.
• [3] Edwards, C. H. (2018). Calculus. Pearson Education.

Keywords

• Integral
• Calculus
• Power rule of integration
• Constant multiple rule of integration
• Integration by substitution
• Integration by parts
  

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Introduction

In our previous article, we evaluated the integral $\int \left(x^3 - 3x\right) \, dx$. We broke down the integral into two separate integrals, evaluated each one using the power rule and constant multiple rule of integration, and combined the results to get the final answer. In this article, we will answer some frequently asked questions related to evaluating this integral.

Q: What is the power rule of integration?

A: The power rule of integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. This rule is used to integrate functions of the form $x^n$, where $n$ is a constant.

Q: What is the constant multiple rule of integration?

A: The constant multiple rule of integration states that $\int af(x) \, dx = a\int f(x) \, dx$. This rule is used to integrate functions of the form $af(x)$, where $a$ is a constant.

Q: How do I evaluate the integral of $x^3$?

A: To evaluate the integral of $x^3$, we can use the power rule of integration. The integral of $x^3$ is $\frac{x^4}{4} + C$.

Q: How do I evaluate the integral of $-3x$?

A: To evaluate the integral of $-3x$, we can use the constant multiple rule of integration. The integral of $-3x$ is $-3\int x \, dx$. We can then evaluate the integral of $x$ using the power rule of integration.

Q: What is the final answer to the integral $\int \left(x^3 - 3x\right) \, dx$?

A: The final answer to the integral $\int \left(x^3 - 3x\right) \, dx$ is $\frac{x^4}{4} - \frac{3x^2}{2} + C$.

Q: What is the constant of integration $C$?

A: The constant of integration $C$ is a constant that is added to the final answer to the integral. It represents the constant of integration and is usually denoted by the letter $C$.

Q: How do I apply the power rule and constant multiple rule of integration to evaluate the integral $\int \left(x^3 - 3x\right) \, dx$?

A: To apply the power rule and constant multiple rule of integration to evaluate the integral $\int \left(x^3 - 3x\right) \, dx$, we can break down the integral into two separate integrals: $\int x^3 \, dx$ and $\int -3x \, dx$. We can then evaluate each integral using the power rule and constant multiple rule of integration, and combine the results to get the final answer.

Q: What are some common mistakes to avoid when evaluating the integral $\int \left(x^3 - 3x\right) \, dx$?

A: Some common mistakes to avoid when evaluating the integral $\int \left(x^3 - 3x\right) \, dx$ include:

• Forgetting to break down the integral into two separate integrals
• Forgetting to apply the power rule and constant multiple rule of integration
• Forgetting to combine the results of the two integrals
• Forgetting to add the constant of integration $C$ to the final answer

Conclusion

In this article, we answered some frequently asked questions related to evaluating the integral $\int \left(x^3 - 3x\right) \, dx$. We covered topics such as the power rule and constant multiple rule of integration, how to evaluate the integral of $x^3$ and $-3x$, and common mistakes to avoid when evaluating the integral.

Final Answer

The final answer to the integral $\int \left(x^3 - 3x\right) \, dx$ is $\boxed{\frac{x^4}{4} - \frac{3x^2}{2} + C}$.

Related Topics

• Power rule of integration
• Constant multiple rule of integration
• Integration by substitution
• Integration by parts

References

• [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
• [2] Anton, H. (2017). Calculus: Early Transcendentals. John Wiley & Sons.
• [3] Edwards, C. H. (2018). Calculus. Pearson Education.

Keywords

• Integral
• Calculus
• Power rule of integration
• Constant multiple rule of integration
• Integration by substitution
• Integration by parts