Find the antiderivative of \( f(x) = 5x^4 - 3x^2 + 7 \)

To find the antiderivative of the function \( f(x) = 5x^4 - 3x^2 + 7 \), we integrate each term separately. The antiderivative is: \[ F(x) = \int (5x^4 - 3x^2 + 7) \, dx \] \[ = \frac{5}{5}x^{5} - \frac{3}{3}x^{3} + 7x + C \] \[ = x^{5} - x^{3} + 7x + C \] where \( C \) is the constant of integration. So, the antiderivative \( F(x) \) is: \[ F(x) = x^{5} - x^{3} + 7x + C \] Now, let's dive into some fun facts about antiderivatives and their applications! The concept of antiderivatives dates back to ancient mathematicians, but it gained significant structure during the Enlightenment with the work of Isaac Newton and Gottfried Wilhelm Leibniz, who independently developed calculus in the 17th century. Their work laid down the foundation for understanding integrals and derivatives, transforming mathematics and sciences forever. In real-world scenarios, antiderivatives are essential in physics and engineering. For example, if you know the velocity of an object as a function of time, integrating that function gives you the object's position over time. This means that when you’re calculating how far a rocket has traveled, the antiderivative of its speed function is your best friend!