Интеграл sqrt(2-x^2) (dx)

Вы ввели

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  1               
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 |     ________   
 |    /      2    
 |  \/  2 - x   dx
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$$\int_{0}^{1} \sqrt{- x^{2} + 2}\, dx$$

Подробное решение

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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:

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

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Ответ:

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

Ответ

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  1                        
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 |     ________            
 |    /      2       1   pi
 |  \/  2 - x   dx = - + --
 |                   2   4 
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$${{\pi+2}\over{4}}$$

Численный ответ

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1.28539816339745

Ответ (Неопределённый)

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  /                                                                                     
 |                                                                                      
 |    ________          //     ________                                                \
 |   /      2           ||    /      2        /    ___\                                |
 | \/  2 - x   dx = C + |<x*\/  2 - x         |x*\/ 2 |         /       ___        ___\|
 |                      ||------------- + asin|-------|  for And\x > -\/ 2 , x < \/ 2 /|
/                       \\      2             \   2   /                                /

$$\int \sqrt{- x^{2} + 2}\, dx = C + \begin{cases} \frac{x}{2} \sqrt{- x^{2} + 2} + \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )} & \text{for}\: x > - \sqrt{2} \wedge x < \sqrt{2} \end{cases}$$

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