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Integral of d{x}:
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Integral of x+
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Integral of (1+cos^2x)/(1+cos2x)
(x^2+y^2)^2
Limit of the function:
(x^2+y^2)^2
Identical expressions
(x^ two +y^ two)^ two
(x squared plus y squared ) squared
(x to the power of two plus y to the power of two) to the power of two
(x2+y2)2
x2+y22
(x²+y²)²
(x to the power of 2+y to the power of 2) to the power of 2
x^2+y^2^2
(x^2+y^2)^2dx
Similar expressions
(x^2-y^2)^2

Integral of (x^2+y^2)^2 dl

The solution

You have entered [src]

 a*cos(t)             
     /                
    |                 
    |             2   
    |    / 2    2\    
    |    \x  + y /  dy
    |                 
   /                  
a*sin(t)

$$\int\limits_{a \sin{\left(t \right)}}^{a \cos{\left(t \right)}} \left(x^{2} + y^{2}\right)^{2}\, dy$$

Integral((x^2 + y^2)^2, (y, a*sin(t), a*cos(t)))

Detail solution

Rewrite the integrand:

$\left(x^{2} + y^{2}\right)^{2} = x^{4} + 2 x^{2} y^{2} + y^{4}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int x^{4}\, dy = x^{4} y$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 x^{2} y^{2}\, dy = 2 x^{2} \int y^{2}\, dy$
  1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int y^{2}\, dy = \frac{y^{3}}{3}$
  So, the result is: $\frac{2 x^{2} y^{3}}{3}$
1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int y^{4}\, dy = \frac{y^{5}}{5}$
The result is: $x^{4} y + \frac{2 x^{2} y^{3}}{3} + \frac{y^{5}}{5}$
Now simplify:

$y \left(x^{4} + \frac{2 x^{2} y^{2}}{3} + \frac{y^{4}}{5}\right)$
Add the constant of integration:

$y \left(x^{4} + \frac{2 x^{2} y^{2}}{3} + \frac{y^{4}}{5}\right)+ \mathrm{constant}$

The answer is:

$y \left(x^{4} + \frac{2 x^{2} y^{2}}{3} + \frac{y^{4}}{5}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                        
 |          2           5             2  3
 | / 2    2\           y       4   2*x *y 
 | \x  + y /  dy = C + -- + y*x  + -------
 |                     5              3   
/

$$\int \left(x^{2} + y^{2}\right)^{2}\, dy = C + x^{4} y + \frac{2 x^{2} y^{3}}{3} + \frac{y^{5}}{5}$$

Simplify

The answer [src]

   5    5       5    5                                     3  2    3         3  2    3   
  a *sin (t)   a *cos (t)      4             4          2*a *x *sin (t)   2*a *x *cos (t)
- ---------- + ---------- + a*x *cos(t) - a*x *sin(t) - --------------- + ---------------
      5            5                                           3                 3

$$- \frac{a^{5} \sin^{5}{\left(t \right)}}{5} + \frac{a^{5} \cos^{5}{\left(t \right)}}{5} - \frac{2 a^{3} x^{2} \sin^{3}{\left(t \right)}}{3} + \frac{2 a^{3} x^{2} \cos^{3}{\left(t \right)}}{3} - a x^{4} \sin{\left(t \right)} + a x^{4} \cos{\left(t \right)}$$

   5    5       5    5                                     3  2    3         3  2    3   
  a *sin (t)   a *cos (t)      4             4          2*a *x *sin (t)   2*a *x *cos (t)
- ---------- + ---------- + a*x *cos(t) - a*x *sin(t) - --------------- + ---------------
      5            5                                           3                 3

-a^5*sin(t)^5/5 + a^5*cos(t)^5/5 + a*x^4*cos(t) - a*x^4*sin(t) - 2*a^3*x^2*sin(t)^3/3 + 2*a^3*x^2*cos(t)^3/3

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

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Identical expressions

Similar expressions

Integral of (x^2+y^2)^2 dl

Limits of integration:

The graph:

Piecewise:

The solution