i4 \( \int 3 x^{4} d x \) 7) \( \int \frac{2}{\sqrt{x}} \cdot x \) 4) \( \left\{y^{3}\left(2 y^{2}-5\right) d\left\{\int x^{1 / 2}(x)\right.\right. \) 47) \( \left\{\left(9 x^{2}-3 z-6 x-1\right) d r\right. \) \( 31 \int 2 x^{3} d x \) 8) \( \int \frac{1}{\sqrt{3}} d y \) v) \( \int \frac{1}{x^{2}} d t \) 1) \( \int 6 t^{?} \sqrt{t} d t \) 15) \( \int\left(8 x^{4}+42^{3}-6 x^{2}-7 x+5\right) d x \) 9) \( \int \frac{1}{x} d t \) (0) \( \{17 x \sqrt[3]{8} d x \) 16) \( \int f(x+1) d x \) (1) \( \int 5 u^{2} d u \) 11) \( \left\{\left(4 x^{3}+2^{2}\right) d x\right. \) (7) \( \left.\int \sqrt{x}-\frac{1}{\sqrt{x}}\right) d x \) 6) \( \int\left(x^{2} \sqrt{x} d x\right. \) (2) \( \int\left(30^{0}-2 u^{0}\right) d \) 18) \( \int\left(3-\frac{1}{x^{7}}+\frac{1}{x^{2}}\right) d x \)
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Imagine the thrill of discovering that integrals have been studied for centuries, tracing back to the ancient Greeks and later advanced by prominent figures like Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Their work laid the foundations for integral calculus, a mathematical marvel that helps us calculate areas under curves, volumes of solids, and so much more! Now, picture yourself using these integrals in real-life scenarios! Whether you're calculating the trajectory of a roller coaster, determining the distance traveled by a car over time, or even optimizing a business's profits, the practical applications of integration are vast. It’s not just numbers on a page; it’s about unlocking the secrets of the universe through mathematics!
