Evaluate The Integral:$\int\left(5 E^x+\frac{3}{x^2}\right) \, D X \quad (x \neq 0$\]

**Evaluating the Integral: A Step-by-Step Guide** =====================================================

Introduction

In this article, we will evaluate the given integral: $\int\left(5 e^x+\frac{3}{x^2}\right) \, d x \quad (x \neq 0)$. This integral involves two separate functions: $5 e^x$ and $\frac{3}{x^2}$. We will use the properties of integration to break down the integral into two separate integrals, and then evaluate each one separately.

Step 1: Break Down the Integral

The given integral can be broken down into two separate integrals:

$$\int\left(5 e^x+\frac{3}{x^2}\right) \, d x = \int 5 e^x \, d x + \int \frac{3}{x^2} \, d x$$

Step 2: Evaluate the First Integral

The first integral is $\int 5 e^x \, d x$. This is a basic integral that can be evaluated using the formula for the integral of $e^x$, which is $\int e^x \, d x = e^x + C$. Since the integral is multiplied by a constant, $5$, we can simply multiply the result by $5$:

$$\int 5 e^x \, d x = 5 \int e^x \, d x = 5 e^x + C$$

Step 3: Evaluate the Second Integral

The second integral is $\int \frac{3}{x^2} \, d x$. This is a basic integral that can be evaluated using the formula for the integral of $\frac{1}{x^2}$, which is $\int \frac{1}{x^2} \, d x = -\frac{1}{x} + C$. Since the integral is multiplied by a constant, $3$, we can simply multiply the result by $3$:

$$\int \frac{3}{x^2} \, d x = 3 \int \frac{1}{x^2} \, d x = -\frac{3}{x} + C$$

Step 4: Combine the Results

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

$$\int\left(5 e^x+\frac{3}{x^2}\right) \, d x = 5 e^x - \frac{3}{x} + C$$

Conclusion

In this article, we evaluated the given integral using the properties of integration. We broke down the integral into two separate integrals, and then evaluated each one separately. The final answer is $5 e^x - \frac{3}{x} + C$.

Q&A

Q: What is the integral of $5 e^x$?

A: The integral of $5 e^x$ is $5 e^x + C$.

Q: What is the integral of $\frac{3}{x^2}$?

A: The integral of $\frac{3}{x^2}$ is $-\frac{3}{x} + C$.

Q: How do I evaluate the integral of a function multiplied by a constant?

A: To evaluate the integral of a function multiplied by a constant, you can simply multiply the result of the integral by the constant.

Q: What is the final answer to the given integral?

A: The final answer to the given integral is $5 e^x - \frac{3}{x} + C$.

Q: What is the condition for the integral to be valid?

A: The integral is valid for $x \neq 0$.

Q: Can I use the same method to evaluate other integrals?

A: Yes, you can use the same method to evaluate other integrals that involve breaking down the integral into separate integrals and then evaluating each one separately.

Q: What is the importance of evaluating integrals?

A: Evaluating integrals is important in many fields, including physics, engineering, and economics, where it is used to solve problems and make predictions.

Q: Can I use a calculator to evaluate integrals?

A: Yes, you can use a calculator to evaluate integrals, but it is also important to understand the underlying math and be able to evaluate integrals by hand.