Интеграл (9-x^2)^(1/2) (dx)

  1               
  /               
 |                
 |     ________   
 |    /      2    
 |  \/  9 - x   dx
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/                 
0                 

TrigSubstitutionRule(theta=_theta, func=3*sin(_theta), rewritten=9*cos(_theta)**2, substep=ConstantTimesRule(constant=9, 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=9*cos(_theta)**2, symbol=_theta), restriction=And(x < 3, x > -3), context=sqrt(-x**2 + 9), symbol=x)

1. Добавляем постоянную интегрирования:

   $\begin{cases} \frac{x}{2} \sqrt{- x^{2} + 9} + \frac{9}{2} \operatorname{asin}{\left (\frac{x}{3} \right )} & \text{for}\: x > -3 \wedge x < 3 \end{cases}+ \mathrm{constant}$

Ответ:

$\begin{cases} \frac{x}{2} \sqrt{- x^{2} + 9} + \frac{9}{2} \operatorname{asin}{\left (\frac{x}{3} \right )} & \text{for}\: x > -3 \wedge x < 3 \end{cases}+ \mathrm{constant}$

  1                                     
  /                                     
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 |     ________                         
 |    /      2         ___   9*asin(1/3)
 |  \/  9 - x   dx = \/ 2  + -----------
 |                                2     
/                                       
0                                       

2.94347965491664

  /                                                                         
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 |    ________          //      /x\        ________                        \
 |   /      2           ||9*asin|-|       /      2                         |
 | \/  9 - x   dx = C + |<      \3/   x*\/  9 - x                          |
 |                      ||--------- + -------------  for And(x > -3, x < 3)|
/                       \\    2             2                              /