Oblicz całkę

[mmigi]

\(\displaystyle{ \int \frac{1}{sinx}dx}\)

[soku11]

\(\displaystyle{ \mathcal{I}=\int \frac{\sin x}{\sin ^2x}\mbox{d}x=
t \frac{\sin x}{1-\cos ^2x}\mbox{d}x\\
\cos x=t\\
\sin x\mbox{d}x=-\mbox{d}t\\
\mathcal{I}=\int \frac{\mbox{d}t}{t^2-1}=
\mathcal{I}=\int \frac{\mbox{d}t}{(t-1)(t+1)}=\ldots}\)

Dalej na ulamki i wiadomo POZDRO

[luka52]

Albo prościej:
\(\displaystyle{ \frac{1}{\sin x} = \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} = \frac{\frac{1}{2} \frac{1}{\cos^2 \frac{x}{2}}}{\tan \frac{x}{2}} = \left( \ln \left| \tan \frac{x}{2} \right| \right)'}\)