Задача №1384
Условие
Найти интеграл \(\int\tg^4{x}dx\).
Решение
\[
\int\tg^4{x}dx
=\int\frac{\sin^4x}{\cos^4x}dx
=\int\frac{\left(1-\cos^2x\right)^2}{\cos^4x}dx=\\
=\int\frac{1-2\cos^2x+\cos^4x}{\cos^4x}dx
=\int\left(\frac{1}{\cos^4x}-2\cdot\frac{1}{\cos^2x}+1\right)dx=\\
=\int\left(\frac{1}{\cos^2x}-2\right)\cdot\frac{dx}{\cos^2x}+\int{dx}
=\int\left(\tg^2{x}-1\right)d(\tg{x})+\int{dx}
=\frac{\tg^3x}{3}-\tg{x}+x+C
\]
Ответ:
\(\frac{\tg^3x}{3}-\tg{x}+x+C\)