Интеграл sqrt(2-x^2) (dx)

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TrigSubstitutionRule(theta=_theta, func=sqrt(2)*sin(_theta), rewritten=2*cos(_theta)**2, substep=ConstantTimesRule(constant=2, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=2*cos(_theta)**2, symbol=_theta), restriction=And(x < sqrt(2), x > -sqrt(2)), context=sqrt(-x**2 + 2), symbol=x)

1. Добавляем постоянную интегрирования:

   $\begin{cases} \frac{x}{2} \sqrt{- x^{2} + 2} + \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )} & \text{for}\: x > - \sqrt{2} \wedge x < \sqrt{2} \end{cases}+ \mathrm{constant}$

Ответ:

$\begin{cases} \frac{x}{2} \sqrt{- x^{2} + 2} + \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )} & \text{for}\: x > - \sqrt{2} \wedge x < \sqrt{2} \end{cases}+ \mathrm{constant}$

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1.28539816339745

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