Задача №1626

\[ \int\frac{dx}{\ch^2{x}} =\int\frac{dx}{\left(\frac{e^x+e^{-x}}{2}\right)^2} =4\cdot\int\frac{dx}{\left(e^x+e^{-x}\right)^2} =4\cdot\int\frac{e^{2x}dx}{\left(e^{2x}+1\right)^2} =2\cdot\int\frac{d\left(e^{2x}+1\right)}{\left(e^{2x}+1\right)^2}=\\ =\frac{-2}{e^{2x}+1}+C =\frac{-2e^{-x}}{e^{x}+e^{-x}}+C =\frac{e^{x}-e^{-x}-\left(e^x+e^{-x}\right)}{e^{x}+e^{-x}}+C =\frac{e^x-e^{-x}}{e^x+e^{-x}}-1+C =\th{x}+C \]