Задача 69591 ...
Условие
математика ВУЗ
101
Решение
★
[m] ∫ \frac{xdx}{\sqrt{1-x^2}}[/m]
[m]d(1-x^2)=(1-x^2)`dx=-2xdx[/m] ⇒ [m]xdx=-\frac{1}{2}d(1-x^2)[/m]
[m] ∫ \frac{xdx}{\sqrt{1-x^2}}=-\frac{1}{2} ∫\frac{d(1-x^2)}{\sqrt{1-x^2}}=-\frac{1}{2}\cdot 2\sqrt{1-x^2} [/m]
[m] ∫^{\frac{1}{2}}_{0} \frac{xdx}{\sqrt{1-x^2}}=(-\sqrt{1-x^2})|^{\frac{1}{2}}_{0}=-\sqrt{1-(\frac{1}{2})^2}+\sqrt{1-0^2}=1-\frac{\sqrt{3}}{2}[/m]