Mister Exam

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Identical expressions
(x^ two + two y)d*y+(2x+y^2)d*y
(x squared plus 2y)d multiply by y plus (2x plus y squared )d multiply by y
(x to the power of two plus two y)d multiply by y plus (2x plus y squared )d multiply by y
(x2+2y)d*y+(2x+y2)d*y
x2+2yd*y+2x+y2d*y
(x²+2y)d*y+(2x+y²)d*y
(x to the power of 2+2y)d*y+(2x+y to the power of 2)d*y
(x^2+2y)dy+(2x+y^2)dy
(x2+2y)dy+(2x+y2)dy
x2+2ydy+2x+y2dy
x^2+2ydy+2x+y^2dy
(x^2+2y)d*y+(2x+y^2)d*ydx
Similar expressions
(x^2+2y)d*y-(2x+y^2)d*y
(x^2+2y)d*y+(2x-y^2)d*y
(x^2-2y)d*y+(2x+y^2)d*y

Solving integrals/ (x^2+2y)d*y+(2x+y^2)d*y

Integral of (x^2+2y)dy+(2x+y^2)dy dx

The solution

You have entered [src]

 oo                                     
  /                                     
 |                                      
 |  // 2      \       /       2\    \   
 |  \\x  + 2*y/*d*y + \2*x + y /*d*y/ dx
 |                                      
/                                       
2

$$\int\limits_{2}^{\infty} \left(y d \left(2 x + y^{2}\right) + y d \left(x^{2} + 2 y\right)\right)\, dx$$

Integral(((x^2 + 2*y)*d)*y + ((2*x + y^2)*d)*y, (x, 2, oo))

Detail solution

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

$\frac{d x y \left(x^{2} + 3 x + 3 y^{2} + 6 y\right)}{3}$
Add the constant of integration:

$\frac{d x y \left(x^{2} + 3 x + 3 y^{2} + 6 y\right)}{3}+ \mathrm{constant}$

The answer is:

$\frac{d x y \left(x^{2} + 3 x + 3 y^{2} + 6 y\right)}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                             
 |                                                                  / 3        \
 | // 2      \       /       2\    \              / 2      2\       |x         |
 | \\x  + 2*y/*d*y + \2*x + y /*d*y/ dx = C + d*y*\x  + x*y / + d*y*|-- + 2*x*y|
 |                                                                  \3         /
/

$$\int \left(y d \left(2 x + y^{2}\right) + y d \left(x^{2} + 2 y\right)\right)\, dx = C + d y \left(x^{2} + x y^{2}\right) + d y \left(\frac{x^{3}}{3} + 2 x y\right)$$

Simplify

The answer [src]

                    2        3   20*d*y
oo*sign(d*y) - 4*d*y  - 2*d*y  - ------
                                   3

$$- 2 d y^{3} - 4 d y^{2} - \frac{20 d y}{3} + \infty \operatorname{sign}{\left(d y \right)}$$

                    2        3   20*d*y
oo*sign(d*y) - 4*d*y  - 2*d*y  - ------
                                   3

$$- 2 d y^{3} - 4 d y^{2} - \frac{20 d y}{3} + \infty \operatorname{sign}{\left(d y \right)}$$

oo*sign(d*y) - 4*d*y^2 - 2*d*y^3 - 20*d*y/3

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

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Identical expressions

Similar expressions

Integral of (x^2+2y)dy+(2x+y^2)dy dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (x^2+2y)d*y+(2x+y^2)d*y dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (x^2+2y)dy+(2x+y^2)dy dx