Step 1 :The question is asking for the indefinite integral of the function \(3x^2 + 8x + 9\). The indefinite integral of a function is the antiderivative of that function. The antiderivative of a function is the function whose derivative is the original function.
Step 2 :The antiderivative of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\), and the antiderivative of a constant is the constant times \(x\).
Step 3 :So, to find the antiderivative of \(3x^2 + 8x + 9\), we can apply these rules to each term separately.
Step 4 :The antiderivative of the function \(3x^2 + 8x + 9\) is \(x^3 + 4x^2 + 9x\).
Step 5 :However, the general form of an indefinite integral also includes a constant of integration, typically denoted as \(C\). This is because the derivative of a constant is zero, so when taking the antiderivative, we don't know what constant may have been there originally.
Step 6 :Therefore, the final answer should be \(x^3 + 4x^2 + 9x + C\).
Step 7 :Final Answer: \(\boxed{x^3 + 4x^2 + 9x + C}\)