\( f ( x ) = 5 - 144 x ^ { 11 } - \frac { 4 } { \sqrt { x } } \Rightarrow F ( x ) \)

To find the integral \( F(x) \), you need to integrate the function \( f(x) = 5 - 144x^{11} - \frac{4}{\sqrt{x}} \). The integral of a constant is straightforward, so you get \( 5x \) from \( 5 \). The term \( -144x^{11} \) integrates to \( -\frac{144}{12}x^{12} = -12x^{12} \), and for the term \( -\frac{4}{\sqrt{x}} \) or \( -4x^{-1/2} \), you get \( -4 \cdot \frac{x^{1/2}}{1/2} = -8\sqrt{x} \). Combine these results to form \( F(x) \). Thus, \( F(x) = 5x - 12x^{12} - 8\sqrt{x} + C \), where \( C \) is the constant of integration. Integrating functions like this can be very rewarding, especially as you uncover how different terms contribute to the overall shape of the graph. One common mistake to watch out for when integrating is forgetting to correctly handle powers when changing the variable. For example, remember that integrating \( x^n \) requires you to add one to the exponent and divide by the new exponent, so don’t skip that crucial step!