Интеграл sqrt(7-x^2) (dx)

1. TrigSubstitutionRule(theta=_theta, func=sqrt(7)*sin(_theta), rewritten=7*cos(_theta)**2, substep=ConstantTimesRule(constant=7, 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=7*cos(_theta)**2, symbol=_theta), restriction=And(x < sqrt(7), x > -sqrt(7)), context=sqrt(-x**2 + 7), symbol=x)$$

2. Добавляем постоянную интегрирования:

   $$
   begin{cases} frac{x}{2} sqrt{- x^{2} + 7} + frac{7}{2} {asin}{left (frac{sqrt{7} x}{7} right )} & text{for}: x > – sqrt{7} wedge x < sqrt{7} end{cases}+ mathrm{constant}

Ответ:

begin{cases} frac{x}{2} sqrt{- x^{2} + 7} + frac{7}{2} {asin}{left (frac{sqrt{7} x}{7} right )} & text{for}: x > – sqrt{7} wedge x < sqrt{7} end{cases}+ mathrm{constant}

1
/ / ___
| |/ 7 |
| ________ ___ 7*asin|—–|
| / 2 / 6 7 /
| / 7 – x dx = —– + ————-
| 2 2
/
0

2.58133327468472

/
| // / ___
| ________ || |x*/ 7 | ________ |
| / 2 ||7*asin|——-| / 2 |
| / 7 – x dx = C + |< 7 / x*/ 7 - x / ___ ___| | ||--------------- + ------------- for Andx > -/ 7 , x < / 7 /| / || 2 2 | /