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Solving integrals/ x^2+y^2+1

Integral of x^2+y^2+1 dy

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
       4                     
       /                     
      |                      
      |      / 2    2    \   
      |      \x  + y  + 1/ dy
      |                      
     /                       
   _________                 
  /       2                  
\/  16 - x
∫16−x24((x2+y2)+1) dy\int\limits_{\sqrt{16 - x^{2}}}^{4} \left(\left(x^{2} + y^{2}\right) + 1\right)\, dy 
Integral(x^2 + y^2 + 1, (y, sqrt(16 - x^2), 4))
Detail solution

  1. Integrate term-by-term:

    1. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

        ∫x2 dy=x2y\int x^{2}\, dy = x^{2} y

      1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫y2 dy=y33\int y^{2}\, dy = \frac{y^{3}}{3}

      The result is: x2y+y33x^{2} y + \frac{y^{3}}{3}

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 dy=y\int 1\, dy = y

    The result is: x2y+y33+yx^{2} y + \frac{y^{3}}{3} + y

  2. Now simplify:

    y(x2+y23+1)y \left(x^{2} + \frac{y^{2}}{3} + 1\right)

  3. Add the constant of integration:

    y(x2+y23+1)+constanty \left(x^{2} + \frac{y^{2}}{3} + 1\right)+ \mathrm{constant}

The answer is:

y(x2+y23+1)+constanty \left(x^{2} + \frac{y^{2}}{3} + 1\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
 |                             3       
 | / 2    2    \              y       2
 | \x  + y  + 1/ dy = C + y + -- + y*x 
 |                            3        
/
∫((x2+y2)+1) dy=C+x2y+y33+y\int \left(\left(x^{2} + y^{2}\right) + 1\right)\, dy = C + x^{2} y + \frac{y^{3}}{3} + y
Simplify
The answer [src] 
                     3/2                        
            /      2\         _________         
76      2   \16 - x /        /       2  /     2\
-- + 4*x  - ------------ - \/  16 - x  *\1 + x /
3                3
4x2−(16−x2)323−16−x2(x2+1)+7634 x^{2} - \frac{\left(16 - x^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{16 - x^{2}} \left(x^{2} + 1\right) + \frac{76}{3} 
=
= 
                     3/2                        
            /      2\         _________         
76      2   \16 - x /        /       2  /     2\
-- + 4*x  - ------------ - \/  16 - x  *\1 + x /
3                3
4x2−(16−x2)323−16−x2(x2+1)+7634 x^{2} - \frac{\left(16 - x^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{16 - x^{2}} \left(x^{2} + 1\right) + \frac{76}{3} 
76/3 + 4*x^2 - (16 - x^2)^(3/2)/3 - sqrt(16 - x^2)*(1 + x^2)

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

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