Simplify $\int_{-1}^0 (3x + 2)^2 , Dx$.

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Introduction

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The given problem is to simplify the definite integral of the function $(3x + 2)^2$ from $-1$ to $0$. This involves evaluating the area under the curve of the function within the specified interval. To simplify the integral, we will use various techniques from calculus, including substitution and expansion of the squared function.

Expanding the Squared Function

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The first step in simplifying the integral is to expand the squared function $(3x + 2)^2$. This can be done using the formula $(a + b)^2 = a^2 + 2ab + b^2$. In this case, $a = 3x$ and $b = 2$. Therefore, we can expand the function as follows:

$$(3x + 2)^2 = (3x)^2 + 2(3x)(2) + 2^2$$

$$= 9x^2 + 12x + 4$$

Evaluating the Integral

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Now that we have expanded the squared function, we can evaluate the integral. The integral of a polynomial function can be evaluated using the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. We will apply this rule to each term in the expanded function.

$$\int (9x^2 + 12x + 4) \, dx = \int 9x^2 \, dx + \int 12x \, dx + \int 4 \, dx$$

$$= \frac{9x^3}{3} + \frac{12x^2}{2} + 4x + C$$

$$= 3x^3 + 6x^2 + 4x + C$$

Applying the Fundamental Theorem of Calculus

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The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$. In this case, we have found the antiderivative of the function $(3x + 2)^2$ to be $3x^3 + 6x^2 + 4x + C$. We can now apply the fundamental theorem of calculus to evaluate the definite integral.

$$\int_{-1}^0 (3x + 2)^2 \, dx = \left[ 3x^3 + 6x^2 + 4x \right]_{-1}^0$$

$$= \left( 3(0)^3 + 6(0)^2 + 4(0) \right) - \left( 3(-1)^3 + 6(-1)^2 + 4(-1) \right)$$

$$= 0 - (-3 + 6 - 4)$$

$$= 0 + 7$$

$$= 7$$

Conclusion

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In this article, we have simplified the definite integral of the function $(3x + 2)^2$ from $-1$ to $0$. We first expanded the squared function using the formula $(a + b)^2 = a^2 + 2ab + b^2$. Then, we evaluated the integral using the power rule of integration. Finally, we applied the fundamental theorem of calculus to evaluate the definite integral. The result is $\boxed{7}$.

Step-by-Step Solution

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Here is a step-by-step solution to the problem:

1. Expand the squared function $(3x + 2)^2$ using the formula $(a + b)^2 = a^2 + 2ab + b^2$.
2. Evaluate the integral of the expanded function using the power rule of integration.
3. Apply the fundamental theorem of calculus to evaluate the definite integral.
4. Simplify the result to obtain the final answer.

Frequently Asked Questions

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Q: What is the definition of a definite integral?

A: A definite integral is a mathematical concept that represents the area under a curve of a function within a specified interval.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

Q: How do I evaluate a definite integral?

A: To evaluate a definite integral, you need to find the antiderivative of the function and then apply the fundamental theorem of calculus.

References

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] The Fundamental Theorem of Calculus, Wikipedia

Keywords

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• Definite integral
• Fundamental theorem of calculus
• Power rule of integration
• Antiderivative
• Calculus
• Mathematics
  

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Introduction

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In the previous article, we simplified the definite integral of the function $(3x + 2)^2$ from $-1$ to $0$. In this article, we will answer some frequently asked questions (FAQs) on definite integrals. These questions cover various topics, including the definition of a definite integral, the fundamental theorem of calculus, and how to evaluate a definite integral.

Q: What is the definition of a definite integral?

A: A definite integral is a mathematical concept that represents the area under a curve of a function within a specified interval. It is denoted by the symbol $\int_{a}^{b} f(x) \, dx$ and is equal to the area between the curve of the function and the x-axis within the interval $[a, b]$.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$. This theorem is a fundamental concept in calculus and is used to evaluate definite integrals.

Q: How do I evaluate a definite integral?

A: To evaluate a definite integral, you need to follow these steps:

1. Find the antiderivative of the function.
2. Apply the fundamental theorem of calculus to evaluate the definite integral.
3. Simplify the result to obtain the final answer.

Q: What is the difference between a definite integral and an indefinite integral?

A: A definite integral is a specific value that represents the area under a curve of a function within a specified interval. An indefinite integral, on the other hand, is a function that represents the antiderivative of a function. The indefinite integral is denoted by the symbol $\int f(x) \, dx$ and is used to find the antiderivative of a function.

Q: How do I find the antiderivative of a function?

A: To find the antiderivative of a function, you need to use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. You can also use other integration techniques, such as substitution and integration by parts, to find the antiderivative of a function.

Q: What is the power rule of integration?

A: The power rule of integration is a fundamental concept in calculus that states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. This rule is used to find the antiderivative of a function and is a key concept in evaluating definite integrals.

Q: How do I use the fundamental theorem of calculus to evaluate a definite integral?

A: To use the fundamental theorem of calculus to evaluate a definite integral, you need to follow these steps:

1. Find the antiderivative of the function.
2. Evaluate the antiderivative at the upper and lower limits of integration.
3. Subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit to obtain the final answer.

Q: What are some common mistakes to avoid when evaluating definite integrals?

A: Some common mistakes to avoid when evaluating definite integrals include:

• Forgetting to evaluate the antiderivative at the upper and lower limits of integration.
• Not simplifying the result to obtain the final answer.
• Using the wrong integration technique or formula.

Conclusion

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In this article, we have answered some frequently asked questions (FAQs) on definite integrals. These questions cover various topics, including the definition of a definite integral, the fundamental theorem of calculus, and how to evaluate a definite integral. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of definite integrals.

Step-by-Step Solution

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Here is a step-by-step solution to the problem:

1. Read the problem carefully and understand what is being asked.
2. Identify the function and the interval of integration.
3. Find the antiderivative of the function using the power rule of integration or other integration techniques.
4. Evaluate the antiderivative at the upper and lower limits of integration.
5. Subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit to obtain the final answer.

Frequently Asked Questions

---

Q: What is the definition of a definite integral?

A: A definite integral is a mathematical concept that represents the area under a curve of a function within a specified interval.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

Q: How do I evaluate a definite integral?

A: To evaluate a definite integral, you need to find the antiderivative of the function and then apply the fundamental theorem of calculus.

References

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] The Fundamental Theorem of Calculus, Wikipedia

Keywords

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• Definite integral
• Fundamental theorem of calculus
• Power rule of integration
• Antiderivative
• Calculus
• Mathematics