\( \left. \begin{array} { l } { \int x ^ { 4 } \sqrt { 3 + x ^ { 5 } } d x } \\ { \frac { 1 } { 5 } \int ( 3 + x ^ { 5 } ) ^ { 1 / 2 } } \\ { \frac { 1 } { 5 } \int \frac { 2 } { 3 } ( 3 + x ^ { 5 } ) ^ { \frac { 3 } { 2 } } } \\ { \frac { 1 } { 5 } \cdot \frac { 2 } { 3 } ( 3 + x ^ { 5 } ) ^ { \frac { 3 } { 2 } } } \\ { \frac { 2 } { 15 } ( 3 + x ^ { 5 } ) ^ { \frac { 3 } { 2 } } + c } \end{array} \right. \)

Did you know that integral calculus has roots dating back to the ancient Greeks? Mathematicians like Archimedes used geometric methods to approximate the area under curves, which laid the groundwork for what would evolve into integral calculus! Fast forward to the 17th century, and genius minds like Newton and Leibniz formalized these ideas, leading to the powerful tools we use today for solving complex integrals. In real-world applications, integrals are crucial in fields like physics and engineering! For example, they help compute the area under curves, which can represent quantities like distance, probability, and even the electric charge in a circuit over time. So next time you tackle an integral, remember, you're tapping into a powerful tool that helps engineers design bridges, physicists understand motion, and so much more!