całka z cos do kwadratu

[renatkan]

\(\displaystyle{ \int}\) \(\displaystyle{ \frac{x}{ cos^{2}x }dx}\)

[sea_of_tears]

\(\displaystyle{ \int\frac{x}{cos^2x}dx=
\begin{cases}
u=x & v^{'}=\frac{1}{cos^2x} \\
u^{'}=1 & v=tgx
\end{cases}
\newline
=xtgx -\int 1\cdot tgx dx=
xtgx - t tgx dx=
xtgx -\int \frac{sinx}{cosx}dx=
\newline
=xtgx +\int\frac{-sinx}{cosx}dx=
xtgx+ln|cosx| +c}\)