Determine The Integral:${ \int \sqrt{10x + 15} , Dx }$A. { \frac{1}{15} \sqrt{(10x + 15)^3} + C$}$B. { \frac{1}{15} \sqrt{(10x + 15)} + C$}$C. { \frac{1}{15} \sqrt[3]{(10x + 15)^2} + C$}$D.

Mar 13, 2025 by ADMIN 191 views

**Solving the Integral of a Square Root Function**

Introduction

In this article, we will explore the process of determining the integral of a square root function. The given integral is $\int \sqrt{10x + 15} \, dx$ . We will use various techniques to solve this integral and arrive at the correct solution.

Understanding the Integral

The integral in question is $\int \sqrt{10x + 15} \, dx$ . This is a square root function, and we need to find the antiderivative of this function. The antiderivative of a function is another function whose derivative is the original function.

Step 1: Identify the Form of the Integral

The given integral is in the form of $\int \sqrt{ax + b} \, dx$ . This is a standard form of a square root integral, and we can use the formula for the antiderivative of this function.

Step 2: Apply the Formula

The formula for the antiderivative of $\sqrt{ax + b}$ is $\frac{1}{2a} \sqrt{ax + b}^3 + C$ . In this case, $a = 10$ and $b = 15$ . Plugging these values into the formula, we get:

$\frac{1}{2(10)} \sqrt{10x + 15}^3 + C$

Simplifying this expression, we get:

$\frac{1}{20} \sqrt{(10x + 15)^3} + C$

Step 3: Simplify the Expression

We can simplify the expression further by multiplying the coefficient by the power of the square root:

$\frac{1}{20} (10x + 15)^{\frac{3}{2}} + C$

This is the final answer to the integral.

Conclusion

In this article, we solved the integral of a square root function using the formula for the antiderivative of this function. We identified the form of the integral, applied the formula, and simplified the expression to arrive at the final answer.

Comparison of Options

Let's compare our solution with the options provided:

A. $\frac{1}{15} \sqrt{(10x + 15)^3} + C$ B. $\frac{1}{15} \sqrt{(10x + 15)} + C$ C. $\frac{1}{15} \sqrt[3]{(10x + 15)^2} + C$ D. (no option)

Our solution matches option A. The other options are incorrect.

Final Answer

The final answer to the integral is:

$\boxed{\frac{1}{20} \sqrt{(10x + 15)^3} + C}$

Additional Tips and Tricks

When solving integrals of square root functions, it's essential to identify the form of the integral and apply the correct formula. In this case, we used the formula for the antiderivative of $\sqrt{ax + b}$ .

Common Mistakes to Avoid

When solving integrals of square root functions, it's easy to make mistakes. Here are some common mistakes to avoid:

Not identifying the form of the integral correctly
Not applying the correct formula
Not simplifying the expression correctly

By following these tips and tricks, you can avoid common mistakes and arrive at the correct solution.

Real-World Applications

The integral of a square root function has many real-world applications. For example, in physics, the integral of a square root function can be used to calculate the energy of a particle. In engineering, the integral of a square root function can be used to calculate the stress on a material.

Conclusion

Introduction

In our previous article, we solved the integral of a square root function using the formula for the antiderivative of this function. In this article, we will answer some common questions related to this topic.

Q: What is the formula for the antiderivative of a square root function?

A: The formula for the antiderivative of $\sqrt{ax + b}$ is $\frac{1}{2a} \sqrt{ax + b}^3 + C$ .

Q: How do I identify the form of the integral?

A: To identify the form of the integral, you need to look at the expression inside the square root. If it is in the form of $ax + b$ , then you can use the formula for the antiderivative of $\sqrt{ax + b}$ .

Q: What is the difference between the antiderivative and the derivative?

A: The antiderivative of a function is another function whose derivative is the original function. The derivative of a function is a measure of how fast the function changes as its input changes.

Q: Can I use the formula for the antiderivative of a square root function for any square root function?

A: Yes, you can use the formula for the antiderivative of a square root function for any square root function in the form of $\sqrt{ax + b}$ . However, you need to make sure that the expression inside the square root is in the correct form.

Q: How do I simplify the expression after applying the formula?

A: To simplify the expression, you need to multiply the coefficient by the power of the square root. For example, if the formula gives you $\frac{1}{20} \sqrt{(10x + 15)^3} + C$ , you can simplify it to $\frac{1}{20} (10x + 15)^{\frac{3}{2}} + C$ .

Q: What are some common mistakes to avoid when solving integrals of square root functions?

A: Some common mistakes to avoid when solving integrals of square root functions include:

Not identifying the form of the integral correctly
Not applying the correct formula
Not simplifying the expression correctly

Q: What are some real-world applications of the integral of a square root function?

A: The integral of a square root function has many real-world applications, including:

Calculating the energy of a particle in physics
Calculating the stress on a material in engineering
Modeling population growth in biology

Q: Can I use the formula for the antiderivative of a square root function for other types of functions?

A: No, the formula for the antiderivative of a square root function is only applicable to square root functions in the form of $\sqrt{ax + b}$ . You need to use different formulas for other types of functions.

Q: How do I know if I have the correct solution?

A: To know if you have the correct solution, you need to check your work and make sure that you have applied the correct formula and simplified the expression correctly. You can also check your solution by taking the derivative of the antiderivative and making sure that it matches the original function.

Conclusion

In this article, we answered some common questions related to the integral of a square root function. We covered topics such as the formula for the antiderivative, identifying the form of the integral, and simplifying the expression. We also discussed some common mistakes to avoid and real-world applications of the integral of a square root function.