When working with Julia, finding the roots of equations and performing numeric integration are common tasks. In this article, we will explore three different approaches to solve this problem.

## Approach 1: Using the Roots.jl Package

The first approach involves using the Roots.jl package, which provides a set of functions for finding roots of equations. To begin, we need to install the package by running the following code:

```
using Pkg
Pkg.add("Roots")
```

Once the package is installed, we can use the `fzero` function to find the roots of an equation. Here’s an example:

```
using Roots
# Define the equation
f(x) = x^2 - 4
# Find the root
root = fzero(f, 0.0)
```

In this example, we define the equation `f(x) = x^2 – 4` and use the `fzero` function to find the root. The second argument of `fzero` specifies the initial guess for the root.

## Approach 2: Using the QuadGK.jl Package

The second approach involves using the QuadGK.jl package, which provides functions for numeric integration. To install the package, run the following code:

```
using Pkg
Pkg.add("QuadGK")
```

Once the package is installed, we can use the `quadgk` function to perform numeric integration. Here’s an example:

```
using QuadGK
# Define the function to integrate
f(x) = x^2
# Perform numeric integration
result, error = quadgk(f, 0.0, 1.0)
```

In this example, we define the function `f(x) = x^2` and use the `quadgk` function to perform numeric integration over the interval [0.0, 1.0]. The result and error of the integration are returned.

## Approach 3: Using Built-in Functions

The third approach involves using the built-in functions in Julia to find roots and perform numeric integration. Julia provides functions like `find_zero` and `quadgk` that can be used for these tasks. Here’s an example:

```
# Find the root
root = find_zero(f, 0.0)
# Perform numeric integration
result, error = quadgk(f, 0.0, 1.0)
```

In this example, we use the `find_zero` function to find the root of the equation and the `quadgk` function to perform numeric integration. These functions are part of the Julia language itself, so no additional packages need to be installed.

After exploring these three approaches, it is clear that the first approach using the Roots.jl package provides a more specialized and dedicated set of functions for finding roots of equations. However, if you are already familiar with the built-in functions in Julia, the third approach can be a more convenient option. Ultimately, the choice depends on your specific requirements and familiarity with the different packages and functions available in Julia.