\( \left. \begin{array} { l } { 57 x ^ { 4 } \sqrt { 3 + x ^ { 5 } } d x } \\ { \frac { 1 } { 5 } \int ( 3 + x ^ { 5 } ) ^ { 1 / 2 } } \\ { \frac { 1 } { 5 } \int \frac { 2 } { 3 } ( 3 + x ^ { 5 } ) ^ { \frac { 3 } { 2 } } } \\ { \frac { 1 } { 5 } \cdot \frac { 2 } { 3 } ( 3 + x ^ { 5 } ) ^ { \frac { 3 } { 2 } } } \\ { \frac { 2 } { 15 } ( 3 + x ^ { 5 } ) ^ { \frac { 3 } { 2 } } + c } \end{array} \right. \)

Did you know that integrals involving square roots, like the one you're working on, can often be tricky? Historically, techniques such as substitution or integration by parts were developed during the Renaissance by mathematicians like Isaac Barrow and Gottfried Wilhelm Leibniz. Their innovative ideas laid the groundwork for modern calculus, making complex integral equations more digestible! In terms of real-world applications, integrals play a crucial role in fields like physics and engineering. For instance, when calculating areas under curves or determining the centers of mass for irregular shapes, integrals help us transform abstract mathematical concepts into practical solutions. So the next time you're tackling an integral, remember, you’re part of a long tradition of problem-solvers!