is created by David Witten, a mathematics and computer science student at Vanderbilt University. For more information, see the "About" page.

Finding the Integral of sec(x)

So, lets say you want to find the integral of sec(x).

MathJax TeX Test Page $$\int\mathrm{\sec{(x)}}\, \mathrm{d}x$$ Initially, this seems really hard, and it kinda is. You have to multiple $\sec(x)$ by $\frac{\sec(x) + \tan(x)}{\sec(x) + \tan(x)}$. Now, you have: $$\int\mathrm{\sec{(x)}\frac{\sec{(x)} + \tan{(x)}}{\sec{(x)} + \tan{(x)}}}\, \mathrm{d}x$$ $$=\int\mathrm{\frac{\sec^2{(x)} + \sec{(x)}\tan{(x)}}{\sec{(x)} + \tan{(x)}}}\, \mathrm{d}x$$
MathJax TeX Test Page Note that the numerator is the derviative of $\tan{(x)} + \sec{(x)}$, so you can do u-substitution. $$u = \sec{(x)} + \tan{(x)}, du = (\sec^2{(x)} + \sec{(x)}\tan{(x)})dx$$
$$\int\mathrm{\frac{1}{u}}\, d{u}$$ $$= \ln{|u|} + C$$ $$= \ln{|\sec{(x)} + \tan{(x)}|} + C$$

David Witten

Proving the limit of a quadratic

Area of Surface of Revolution