Задача №1273
Условие
Найти интеграл \(\int\frac{x^4dx}{\sqrt{4+x^5}}\).
Решение
\[
\int\frac{x^4dx}{\sqrt{4+x^5}}
=\left[\begin{aligned}& d\left(4+x^5\right)=5x^4dx;\\& x^4dx=\frac{1}{5}d\left(4+x^5\right).\end{aligned}\right]=\\
=\frac{1}{5}\int\left(4+x^5\right)^{-\frac{1}{2}}d\left(4+x^5\right)
=\frac{1}{5}\cdot\frac{\left(4+x^5\right)^{\frac{1}{2}}}{\frac{1}{2}}+C
=\frac{2\sqrt{4+x^5}}{5}+C
\]
Ответ:
\(\frac{2\sqrt{4+x^5}}{5}+C\)