Everyday Math 每日數學 之 Integration #20190529 (+solution)

Let $u=\sqrt{x}$
Then $x=u^2$
$dx=2udu$
$\int \sec^2{\sqrt{x}}dx$
$=\int 2u\sec^2 udu$
$=\int 2ud(\tan u)$
$=2u\tan u-2\int \tan u du$
$=2u\tan u-2\int \frac{\sin u}{\cos u} du$
$=2u\tan u+2\int \frac{1}{\cos u} d(\cos u)$
$=2u\tan u+2\ln \left|\cos u \right| +C$
$=2\sqrt{x}\tan \sqrt{x}+2\ln \left|\cos \sqrt{x} \right| +C$