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

Did you know that the process of integration has a rich history that dates back to ancient civilizations? The foundations of integral calculus were laid by Mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. They not only developed techniques for calculating areas under curves but also introduced concepts such as limits and infinitesimals, which are fundamental to modern calculus! In the real world, integrals are crucial in various fields such as physics, engineering, and economics. For instance, in physics, integrals are used to calculate quantities like displacement from velocity or area under a curve, which represents total energy. In economics, integrals can help determine consumer surplus or market demand over a range of prices, showcasing the power of calculus in modeling and analysis across diverse scenarios!