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sqrt(8-2x-x^2)
Solving integrals/ sqrt(8-2x-x^2)

Integral of sqrt(8-2x-x^2) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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 |  \/  8 - 2*x - x   dx
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0
∫01−x2−2x+8 dx\int\limits_{0}^{1} \sqrt{- x^{2} - 2 x + 8}\, dx 
Integral(sqrt(8 - 2*x - x^2), (x, 0, 1))
Detail solution

    SqrtQuadraticRule(a=8, b=-2, c=-1, context=sqrt(-x**2 - 2*x + 8), symbol=x)

  1. Now simplify:

    (x+1)−x2−2x+82+9asin⁡(x3+13)2\frac{\left(x + 1\right) \sqrt{- x^{2} - 2 x + 8}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)}}{2}

  2. Add the constant of integration:

    (x+1)−x2−2x+82+9asin⁡(x3+13)2+constant\frac{\left(x + 1\right) \sqrt{- x^{2} - 2 x + 8}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)}}{2}+ \mathrm{constant}

The answer is:

(x+1)−x2−2x+82+9asin⁡(x3+13)2+constant\frac{\left(x + 1\right) \sqrt{- x^{2} - 2 x + 8}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} + \frac{1}{3} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                    
 |                                  /1   x\                            
 |    ______________          9*asin|- + -|      ______________        
 |   /            2                 \3   3/     /      2        /1   x\
 | \/  8 - 2*x - x   dx = C + ------------- + \/  8 - x  - 2*x *|- + -|
 |                                  2                           \2   2/
/
x −x2−2 x+82+−x2−2 x+82−9 arcsin⁡(−2 x−26)2{{x\,\sqrt{-x^2-2\,x+8}}\over{2}}+{{\sqrt{-x^2-2\,x+8}}\over{2}}-{{ 9\,\arcsin \left({{-2\,x-2}\over{6}}\right)}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
  ___     ___   9*asin(1/3)   9*asin(2/3)
\/ 5  - \/ 2  - ----------- + -----------
                     2             2
9 arcsin⁡(23)+2 52−9 arcsin⁡(13)+2322{{9\,\arcsin \left({{2}\over{3}}\right)+2\,\sqrt{5}}\over{2}}-{{9\, \arcsin \left({{1}\over{3}}\right)+2^{{{3}\over{2}}}}\over{2}} 
=
= 
  ___     ___   9*asin(1/3)   9*asin(2/3)
\/ 5  - \/ 2  - ----------- + -----------
                     2             2
−9asin⁡(13)2−2+5+9asin⁡(23)2- \frac{9 \operatorname{asin}{\left(\frac{1}{3} \right)}}{2} - \sqrt{2} + \sqrt{5} + \frac{9 \operatorname{asin}{\left(\frac{2}{3} \right)}}{2}
Упростить
Numerical answer [src] 
2.57636277560449
2.57636277560449
The graph
Integral of sqrt(8-2x-x^2) dx

    Use the examples entering the upper and lower limits of integration.

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