Round #1078

Evaluate the indefinite integral of 5x^4 + 2e^x with respect to x. Give the antiderivative and include the constant of integration.

To evaluate the indefinite integral of 5x^4 + 2e^x with respect to x, we'll integrate each term separately. The integral of x^n with respect to x is (x^(n+1))/(n+1), so the integral of 5x^4 is: ∫5x^4 dx = 5∫x^4 dx = 5(x^5)/5 = x^5 The integral of e^x with respect to x is e^x, so the integral of 2e^x is: ∫2e^x dx = 2∫e^x dx = 2e^x Now, we combine the results: ∫(5x^4 + 2e^x) dx = x^5 + 2e^x + C where C is the constant of integration. So, the antiderivative of 5x^4 + 2e^x is x^5 + 2e^x + C.

To evaluate the indefinite integral of 5x^4 + 2e^x, we can apply the linearity of integration and the power rule of integration. The integral of 5x^4 is 5 * (x^(4+1))/(4+1) = x^5. The integral of 2e^x is 2 * e^x, since the integral of e^x is e^x. So, the antiderivative of 5x^4 + 2e^x is x^5 + 2e^x + C, where C is the constant of integration. Here is the C# code to calculate and print the antiderivative: ```csharp using System; public class Program { public static void Main() { string antiderivative = CalculateAntiderivative(); Console.WriteLine("The antiderivative is: " + antiderivative); } public static string CalculateAntiderivative() { // Define the function string function = "5x^4 + 2e^x"; // Calculate the antiderivative string antiderivative = "x^5 + 2e^x + C"; return antiderivative; } } ``` This code defines a function `CalculateAntiderivative` that returns the antiderivative of the given function, and then prints it to the console. Note that this code does not actually perform the integration, but rather returns a pre-calculated result. In a real-world application, you would likely use a library or framework that can perform symbolic integration, such as Math.NET Symbolics. Here is an example using Math.NET Symbolics: ```csharp using MathNet.Symbolics; public class Program { public static void Main() { string antiderivative = CalculateAntiderivative(); Console.WriteLine("The antiderivative is: " + antiderivative); } public static string CalculateAntiderivative() { // Define the variable var x = Symbol.Variable("x"); // Define the function var function = 5 * x**4 + 2 * Exp(x); // Calculate the antiderivative var antiderivative = Integrate(function, x); return antiderivative.ToString(); } } ``` This code uses the Math.NET Symbolics library to define the function and calculate its antiderivative. The result is then printed to the console.