Evaluate The Following Integral:$\int_0^3 4x(x-5) \, Dx$

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Introduction

In this article, we will evaluate the given integral $\int_0^3 4x(x-5) \, dx$. This involves using the fundamental theorem of calculus and various integration techniques to find the value of the integral. We will break down the problem step by step, starting with the basic definition of an integral and then applying the necessary techniques to solve it.

Basic Definition of an Integral

The integral of a function $f(x)$ with respect to $x$ is denoted by $\int f(x) \, dx$. It represents the area under the curve of the function $f(x)$ between two points $a$ and $b$. In mathematical terms, the integral of $f(x)$ from $a$ to $b$ is defined as:

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$.

Evaluating the Integral

To evaluate the given integral $\int_0^3 4x(x-5) \, dx$, we need to find the antiderivative of the function $4x(x-5)$. We can do this by using the product rule of differentiation, which states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.

Applying the product rule to the function $4x(x-5)$, we get:

$$\frac{d}{dx} (4x(x-5)) = 4(x-5) + 4x \cdot 1$$

Simplifying the expression, we get:

$$\frac{d}{dx} (4x(x-5)) = 4x - 20 + 4x$$

$$\frac{d}{dx} (4x(x-5)) = 8x - 20$$

Finding the Antiderivative

To find the antiderivative of the function $4x(x-5)$, we need to integrate the expression $8x - 20$ with respect to $x$. We can do this by using the power rule of integration, which states that if $f(x) = x^n$, then $\int f(x) \, dx = \frac{x^{n+1}}{n+1} + C$.

Applying the power rule to the expression $8x - 20$, we get:

$$\int (8x - 20) \, dx = \frac{8x^2}{2} - 20x + C$$

Simplifying the expression, we get:

$$\int (8x - 20) \, dx = 4x^2 - 20x + C$$

Evaluating the Integral

Now that we have found the antiderivative of the function $4x(x-5)$, we can evaluate the given integral $\int_0^3 4x(x-5) \, dx$. We can do this by applying the fundamental theorem of calculus, which states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$.

Applying the fundamental theorem of calculus to the given integral, we get:

$$\int_0^3 4x(x-5) \, dx = \left[ 4x^2 - 20x \right]_0^3$$

Evaluating the expression at the limits of integration, we get:

$$\int_0^3 4x(x-5) \, dx = \left[ 4(3)^2 - 20(3) \right] - \left[ 4(0)^2 - 20(0) \right]$$

Simplifying the expression, we get:

$$\int_0^3 4x(x-5) \, dx = \left[ 36 - 60 \right] - \left[ 0 - 0 \right]$$

$$\int_0^3 4x(x-5) \, dx = -24$$

Conclusion

In this article, we evaluated the given integral $\int_0^3 4x(x-5) \, dx$. We used the fundamental theorem of calculus and various integration techniques to find the value of the integral. We broke down the problem step by step, starting with the basic definition of an integral and then applying the necessary techniques to solve it. The final answer to the integral is -24.

Final Answer

The final answer to the integral $\int_0^3 4x(x-5) \, dx$ is $\boxed{-24}$.

Related Topics

• Integration techniques
• Fundamental theorem of calculus
• Power rule of integration
• Product rule of differentiation

References

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Artin
• [3] "Calculus: Early Transcendentals" by James Stewart
  

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Introduction

In our previous article, we evaluated the given integral $\int_0^3 4x(x-5) \, dx$. In this article, we will answer some common questions related to the evaluation of this integral. We will cover topics such as integration techniques, fundamental theorem of calculus, and power rule of integration.

Q1: What is the definition of an integral?

A1: The integral of a function $f(x)$ with respect to $x$ is denoted by $\int f(x) \, dx$. It represents the area under the curve of the function $f(x)$ between two points $a$ and $b$.

Q2: How do you evaluate an integral?

A2: To evaluate an integral, you need to find the antiderivative of the function. The antiderivative is a function that, when differentiated, gives the original function. Once you have the antiderivative, you can apply the fundamental theorem of calculus to evaluate the integral.

Q3: What is the fundamental theorem of calculus?

A3: The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$.

Q4: How do you apply the power rule of integration?

A4: The power rule of integration states that if $f(x) = x^n$, then $\int f(x) \, dx = \frac{x^{n+1}}{n+1} + C$. To apply the power rule, you need to identify the power of $x$ in the function and then integrate it.

Q5: What is the product rule of differentiation?

A5: The product rule of differentiation states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$. This rule is used to find the derivative of a product of two functions.

Q6: How do you evaluate the integral $\int_0^3 4x(x-5) \, dx$?

A6: To evaluate the integral $\int_0^3 4x(x-5) \, dx$, you need to find the antiderivative of the function $4x(x-5)$. You can do this by using the product rule of differentiation and then integrating the result. Once you have the antiderivative, you can apply the fundamental theorem of calculus to evaluate the integral.

Q7: What is the final answer to the integral $\int_0^3 4x(x-5) \, dx$?

A7: The final answer to the integral $\int_0^3 4x(x-5) \, dx$ is $\boxed{-24}$.

Q8: What are some related topics to the evaluation of the integral $\int_0^3 4x(x-5) \, dx$?

A8: Some related topics to the evaluation of the integral $\int_0^3 4x(x-5) \, dx$ include integration techniques, fundamental theorem of calculus, power rule of integration, and product rule of differentiation.

Q9: Where can I find more information on the evaluation of integrals?

A9: You can find more information on the evaluation of integrals in textbooks such as "Calculus" by Michael Spivak, "Introduction to Calculus" by Michael Artin, and "Calculus: Early Transcendentals" by James Stewart.

Q10: How do I practice evaluating integrals?

A10: You can practice evaluating integrals by working through examples and exercises in textbooks or online resources. You can also try evaluating integrals on your own using the techniques and formulas discussed in this article.

Conclusion

In this article, we answered some common questions related to the evaluation of the integral $\int_0^3 4x(x-5) \, dx$. We covered topics such as integration techniques, fundamental theorem of calculus, and power rule of integration. We also provided some related topics and resources for further learning.

Final Answer

The final answer to the integral $\int_0^3 4x(x-5) \, dx$ is $\boxed{-24}$.

Related Topics

• Integration techniques
• Fundamental theorem of calculus
• Power rule of integration
• Product rule of differentiation

References

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Artin
• [3] "Calculus: Early Transcendentals" by James Stewart