Round #1001

Compute the indefinite integral of 6x^2 + 4x dx.

To compute the indefinite integral of 6x^2 + 4x dx, we'll integrate each term separately. The integral of x^n is (x^(n+1))/(n+1), so: ∫6x^2 dx = 6∫x^2 dx = 6(x^3)/3 = 2x^3 The integral of x is (x^2)/2, so: ∫4x dx = 4∫x dx = 4(x^2)/2 = 2x^2 Now, we combine the results: ∫(6x^2 + 4x) dx = 2x^3 + 2x^2 + C where C is the constant of integration. So, the indefinite integral of 6x^2 + 4x dx is 2x^3 + 2x^2 + C.

```csharp using System; public class IntegralCalculator { public static string CalculateIndefiniteIntegral(string function) { // Split the function into terms string[] terms = function.Split('+'); // Initialize the result string result = ""; // Iterate over each term foreach (string term in terms) { // Check if the term is a constant if (!term.Contains("x")) { // If the term is a constant, integrate it as x result += term + "x + "; } else { // Extract the coefficient and exponent from the term string coefficient = ""; string exponent = ""; bool hasCoefficient = false; foreach (char c in term) { if (char.IsDigit(c) || c == '-') { coefficient += c; hasCoefficient = true; } else if (c == 'x') { break; } } if (!hasCoefficient) { coefficient = "1"; } if (term.Contains("^")) { exponent = term.Split('^')[1]; } else { exponent = "1"; } // Calculate the new coefficient and exponent double newCoefficient = double.Parse(coefficient) / (double.Parse(exponent) + 1); double newExponent = double.Parse(exponent) + 1; // Add the integrated term to the result result += newCoefficient + "x^" + newExponent + " + "; } } // Remove the extra " + " at the end result = result.Substring(0, result.Length - 3); // Add the constant of integration result += " + C"; return result; } public static void Main() { string function = "6x^2 + 4x"; Console.WriteLine(CalculateIndefiniteIntegral(function)); } } ``` This C# code calculates the indefinite integral of the given function 6x^2 + 4x. It splits the function into terms, integrates each term separately using the power rule of integration, and combines the results. The constant of integration is added at the end. The result is then printed to the console. The output of this code will be: 2x^3 + 2x^2 + C.