Question

Выч несоб-ый интеграл и уст расх

Answer 1

$\int tgx\, dx=\int \frac{sinx}{cosx}\, dx=\int \frac{-d(cosx)}{cosx}=\Big [\; \int \frac{dt}{t}=ln|t|+C\; \Big ]=-ln|cosx|+C\; ;\\\\\\\int\limits^{\pi }_{\frac{\pi}{2}}\, tgx\, dx=\lim\limits _{\varepsilon \to +0}\Big (-ln|cosx|\Big |_{\frac{\pi}{2}+\varepsilon }^{\pi }\Big )=\lim\limits _{\varepsilon \to +0}\Big (-ln|cos\pi |+ln|cos(\frac{\pi}{2}+\varepsilon )|\Big )=\\\\=\lim\limits_{\varepsilon \to +0}\Big (-ln|-1|+ln|-sin\varepsilon |\Big )=\lim\limits_{\varepsilon \to +0}\Big (-ln1+ln(sin\varepsilon )\Big )=\\\\=-0+(-\infty )=-\infty \; \; \; \Rightarrow \; \; \; rasxoditsya$