\( \int \frac { d x } { ( 5 + x ) \cdot \sqrt { x + 1 } } \)

To solve the integral \( \int \frac{dx}{(5 + x) \cdot \sqrt{x + 1}} \), we can use a substitution method. Let's set \( u = \sqrt{x + 1} \). Then, \( x = u^2 - 1 \) and \( dx = 2u \, du \). The integral becomes: \[ \int \frac{2u \, du}{(5 + (u^2 - 1)) \cdot u} = 2 \int \frac{du}{(u^2 + 4) \cdot \sqrt{x + 1}}. \] This transforms our integral into a more manageable form, allowing us to handle it using trigonometric or partial fraction methods. As you progress, remember to do the back-substitution to revert to the original variable at the end. --- An interesting fact about integrals: many problems become much easier with the right substitution! That’s why mathematicians often refer to integration as a "toolbox" of methods. Embracing different techniques can transform a complex integral into something quite straightforward. The history of calculus, particularly integrals, goes back centuries! It was developed independently by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Their revolutionary ideas paved the way for modern mathematics, allowing for all sorts of real-world applications, from physics to engineering. Understanding these roots can deepen your appreciation of the subject!