Задача №1661
Условие
Найти интеграл \(\int\limits_{1/2}^{\sqrt{3}/2}\frac{x^3dx}{\left(\frac{5}{8}-x^4\right)\sqrt{\frac{5}{8}-x^4}}\).
Решение
\[
\int\limits_{1/2}^{\sqrt{3}/2}\frac{x^3dx}{\left(\frac{5}{8}-x^4\right)\sqrt{\frac{5}{8}-x^4}}
=-\frac{1}{4}\cdot\int\limits_{1/2}^{\sqrt{3}/2}\left(\frac{5}{8}-x^4\right)^{-\frac{3}{2}}d\left(\frac{5}{8}-x^4\right)
=\frac{1}{2}\cdot\left.\left(\frac{5}{8}-x^4\right)^{-\frac{1}{2}}\right|_{1/2}^{\sqrt{3}/2}
=\frac{4}{3}.
\]
Ответ:
\(\frac{4}{3}\)