\[\int\limits_0^\pi {\ln \left( {1 + \frac{1}{{\sin x}}} \right)dx} \]

\[\int\limits_0^{\frac{\pi }{2}} {\frac{x}{{\sin x}}dx} \]

\[\int\limits_0^1 {\frac{{\arctan \left( {{x^k}} \right)}}{x}dx} \]

\[\int\limits_0^1 {\frac{{\ln x}}{{1 + {x^2}}}dx} \]

\[\int\limits_0^1 {\int\limits_0^1 {\frac{1}{{1 + {{\left( {xy} \right)}^2}}}dydx} } \]

\[\int\limits_0^{\frac{\pi }{4}} {\ln \left( {\operatorname{tg} x} \right)dx} \]

\[\int\limits_0^{\frac{\pi }{2}} {\int\limits_0^{\frac{\pi }{2}} {\frac{{\sin x\sin y}}{{{{\sin }^2}x + {{\sin }^2}y}}dxdy} } = Catalan.\]

\[\int\limits_0^{ + \infty } {\frac{x}{{{e^x} + {e^{ - x}}}}dx} \]

$$\int\limits_0^{ + \infty } {\frac{1}{{\cosh \sqrt x }}dx} $$

$$\int\limits_0^{ + \infty } {\operatorname{arctg} \left( {{e^{ - x}}} \right)dx} $$