Questions: Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) f(x)=4 x+7 F(x)=

Solution Steps

The given function is \( f(x) = 4x + 7 \). We need to find its most general antiderivative \( F(x) \).

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \). Apply this rule to each term in \( f(x) \):

1. For \( 4x \), the antiderivative is \( \frac{4x^2}{2} = 2x^2 \).
2. For \( 7 \), the antiderivative is \( 7x \).

Combine the antiderivatives of each term and include the constant of integration \( C \):

\[ F(x) = 2x^2 + 7x + C \]

Differentiate \( F(x) \) to ensure it matches the original function \( f(x) \):

\[ F'(x) = \frac{d}{dx}(2x^2 + 7x + C) = 4x + 7 \]

Since \( F'(x) = f(x) \), the antiderivative is correct.

Final Answer