Question

Evaluate: $\int \sqrt{secx}dx$

Answer 1

$\int \sqrt{secx}.dx=\int \frac{1}{\sqrt{cosx}}.dx$

Let $cosx=u$ then $-sinx.dx=du$

$-\sqrt{1-co{s}^{2}x}.dx=du$

$-\sqrt{1-u^2}.dx=du$

$dx=\frac{du}{\sqrt{1-u^2}}$

$=\int -\frac{1}{\sqrt{u(1-u^2)}}du$

$=-\int \frac{1}{\sqrt{cosx(1-co{s}^{2}x)}}.du$

It is an example elliptical integration.