Задача №1936
Условие
Найти интеграл \(\int\sh^4{x}\ch^4{x}dx\).
Решение
\[
\int\sh^4{x}\ch^4{x}dx
=\int\left(\sh{x}\ch{x}\right)^4dx
=\int\left(\frac{\sh{2x}}{2}\right)^4dx
=\frac{1}{16}\cdot\int\left(\sh^2{x}\right)^2dx
=\frac{1}{16}\cdot\int\left(\frac{\ch{4x}-1}{2}\right)^2dx=\\
=\frac{1}{64}\cdot\int\left(\ch^2{4x}-2\ch{4x}+1\right)dx
=\frac{1}{64}\cdot\int\left(\frac{\ch{8x}}{2}-2\ch{4x}+\frac{3}{2}\right)dx
=\frac{\sh{8x}}{1024}-\frac{\sh{4x}}{128}+\frac{3x}{128}+C
\]
Ответ:
\(\frac{\sh{8x}}{1024}-\frac{\sh{4x}}{128}+\frac{3x}{128}+C\)