Интеграл x^2*sqrt(25-x^2) (dx)

$$int_{0}^{1} x^{2} sqrt{- x^{2} + 25}, dx$$

1. TrigSubstitutionRule(theta=_theta, func=5*sin(_theta), rewritten=-625*cos(4*_theta)/8 + 625/8, substep=AddRule(substeps=[ConstantTimesRule(constant=-625/8, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func=’cos’, arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=-625*cos(4*_theta)/8, symbol=_theta), ConstantRule(constant=625/8, context=625/8, symbol=_theta)], context=-625*cos(4*_theta)/8 + 625/8, symbol=_theta), restriction=And(x < 5, x > -5), context=x**2*sqrt(-x**2 + 25), symbol=x)

2. Теперь упростить:

   begin{cases} frac{x}{8} sqrt{- x^{2} + 25} left(2 x^{2} – 25right) + frac{625}{8} {asin}{left (frac{x}{5} right )} & text{for}: x > -5 wedge x < 5 end{cases} $$

3. Добавляем постоянную интегрирования:

   $$
   begin{cases} frac{x}{8} sqrt{- x^{2} + 25} left(2 x^{2} – 25right) + frac{625}{8} {asin}{left (frac{x}{5} right )} & text{for}: x > -5 wedge x < 5 end{cases}+ mathrm{constant}

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Ответ:

begin{cases} frac{x}{8} sqrt{- x^{2} + 25} left(2 x^{2} – 25right) + frac{625}{8} {asin}{left (frac{x}{5} right )} & text{for}: x > -5 wedge x < 5 end{cases}+ mathrm{constant}

1
/
|
| _________ ___
| 2 / 2 23*/ 6 625*asin(1/5)
| x */ 25 – x dx = – ——– + ————-
| 4 8
/
0

$${{625,arcsin left({{1}over{5}}right)-46,sqrt{6}}over{8}}$$

1.64652154074132

/
|
| _________ // /x _________
| 2 / 2 ||625*asin|-| / 2 / 2 |
| x */ 25 – x dx = C + |< 5/ x*/ 25 - x *25 - 2*x / | | ||----------- - -------------------------- for And(x > -5, x < 5)| / 8 8 /

$$-{{x,left(25-x^2right)^{{{3}over{2}}}}over{4}}+{{25,x,sqrt{
25-x^2}}over{8}}+{{625,arcsin left({{x}over{5}}right)}over{8
}}$$