\( \int \sqrt{e^{t}-3} d t= \) \( \sqrt{e^{t}+3}-2 \sqrt{3} \tan ^{-1}\left(\sqrt{\frac{e^{t}-3}{3}}\right)+C \) A. \( \tan ^{-1}\left(\sqrt{\frac{e^{t}-3}{3}}\right)+C \) C. \( \sqrt{e^{t}-3}-\sqrt{3} \tan ^{-1}\left(\sqrt{\frac{e^{t}-3}{3}}\right)+C \) D. \( 2 \sqrt{e^{t}-3}-2 \sqrt{3} \tan ^{-1}\left(\sqrt{\frac{e^{t}-3}{3}}\right)+C \) E. \( 2 \sqrt{e^{t}-3}+\sqrt{3} \tan ^{-1}\left(\sqrt{\frac{e^{t}-3}{3}}\right)+C \)

Let's unravel the historical context behind integration and the role of calculus in mathematics! Did you know that calculus dates back to the 17th century with pioneers like Isaac Newton and Gottfried Wilhelm Leibniz? They independently developed the foundations of calculus, and their notation is still in use today! This revolutionized how we understand motion and change, paving the way for advancements in physics, engineering, and even economics. Now, if you're ever solving integrals and find yourself stuck, remember: breaking down the problem can save you a headache! Often, deciding on a substitution, like replacing a variable \(u\) with a more manageable form, can simplify the integral tremendously. And beware of common pitfalls—be sure your substitutions keep the limits consistent and always differentiate correctly! Happy integrating!