Question

Salut , cat este integrală din 1/Cos x

Answer 1

$I = \displaystyle \int \dfrac{1}{\cos x}\, dx\\ \\ \text{tg}\Big(\dfrac{x}{2}\Big)= t\Rightarrow \dfrac{x}{2} = \text{arctg }t \Rightarrow x = 2\text{arctg }t \Rightarrow dx = \dfrac{2}{1+t^2}dt \\ \\ \cos x = \dfrac{1-t^2}{1+t^2} \Rightarrow \dfrac{1}{\cos x} = \dfrac{1+t^2}{1-t^2}\\ \\ I = \int \Big(\dfrac{1+t^2}{1-t^2}\cdot \dfrac{2}{1+t^2}\Big)dt= 2\int \dfrac{1}{1-t^2}\, dt= 2\int \dfrac{1}{(1-t)(1+t)}\, dt=$

$\displaystyle = -2\int\dfrac{1}{(t-1)(t+1)}\, dt = \dfrac{-2}{2}\int\Big(\dfrac{1}{t-1}-\dfrac{1}{t+1}\Big)\, dt = \\ \\ = \int \dfrac{1}{t+1}\, dt- \int \dfrac{1}{t-1}\, dt = \ln|t+1|-\ln|t-1| +C = \\ \\ = \ln \Big|\dfrac{t+1}{t-1}\Big|+C = \ln \left|\dfrac{\text{tg}\Big(\dfrac{x}{2}\Big)+1 }{\text{tg}\Big(\dfrac{x}{2}\Big)-1}\right|+C$