To solve the integral , we can use a substitution. Let . Then . Rearranging gives us , allowing us to rewrite the integral as:

So, the integral evaluates to .

Did you know that integration techniques like substitution or integration by parts didn’t happen overnight? In fact, they evolved from mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, creating the foundations for calculus as we know it today—imagine people figuring out the area under curves without computers!

If you’re delving deeper into calculus, consider exploring classic texts like “Calculus” by Michael Spivak or “Calculus Made Easy” by Silvanus P. Thompson. These books not only guide through the complexities of integration but also celebrate the elegance of mathematical concepts, making the learning journey both enriching and enjoyable!