Задача 69673 ...
Условие
математика ВУЗ
82
Решение
★
[m]x=\frac{\sqrt{2}}{sint}[/m] ⇒ [m]x^2-2=(\frac{\sqrt{2}}{sint})^2-2=2\cdot (\frac{1}{sin^2t}-1)=2\cdot (\frac{1-sin^2t}{sin^2t})=2\cdot \frac{cos^2t}{sin^2t}=2\cdot ctg^2t[/m]
[m]dx=-\frac{\sqrt{2}}{sin^2t}\cdot cost dt[/m]
[m]= ∫^{\frac{π}{6}} _{\frac{π}{2}}\frac{\sqrt{2}ctgt}{\frac{4}{sin^4t}}(-\frac{\sqrt{2}}{sin^2t})dt=-\frac{1}{2}∫^{\frac{π}{6}} _{\frac{π}{2}}cos^2tsint dt=(\frac{cos^3t}{6})|^{\frac{π}{6}} _{\frac{π}{2}}=\frac{\sqrt{3}}{16}[/m]