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Integral of d{x}:
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Integral of x/(x+1)
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Identical expressions
(three *x*y+y^ two)*d*x+(x^ three + two *x*y)*d*y
(3 multiply by x multiply by y plus y squared ) multiply by d multiply by x plus (x cubed plus 2 multiply by x multiply by y) multiply by d multiply by y
(three multiply by x multiply by y plus y to the power of two) multiply by d multiply by x plus (x to the power of three plus two multiply by x multiply by y) multiply by d multiply by y
(3*x*y+y2)*d*x+(x3+2*x*y)*d*y
3*x*y+y2*d*x+x3+2*x*y*d*y
(3*x*y+y²)*d*x+(x³+2*x*y)*d*y
(3*x*y+y to the power of 2)*d*x+(x to the power of 3+2*x*y)*d*y
(3xy+y^2)dx+(x^3+2xy)dy
(3xy+y2)dx+(x3+2xy)dy
3xy+y2dx+x3+2xydy
3xy+y^2dx+x^3+2xydy
(3*x*y+y^2)*d*x+(x^3+2*x*y)*d*ydx
Similar expressions
(3*x*y-y^2)*d*x+(x^3+2*x*y)*d*y
(3*x*y+y^2)*d*x-(x^3+2*x*y)*d*y
(3*x*y+y^2)*d*x+(x^3-2*x*y)*d*y

Solving integrals/ (3*x*y+y^2)*d*x+(x^3+2*x*y)*d*y

Integral of (3xy+y^2)dx+(x^3+2xy)dy dx

The solution

You have entered [src]

  1                                         
  /                                         
 |                                          
 |  //         2\       / 3        \    \   
 |  \\3*x*y + y /*d*x + \x  + 2*x*y/*d*y/ dx
 |                                          
/                                           
l

$$\int\limits_{l}^{1} \left(x d \left(3 x y + y^{2}\right) + y d \left(x^{3} + 2 x y\right)\right)\, dx$$

Integral((((3*x)*y + y^2)*d)*x + ((x^3 + (2*x)*y)*d)*y, (x, l, 1))

Detail solution

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

$\frac{d x^{2} y \left(x^{2} + 4 x + 6 y\right)}{4}$
Add the constant of integration:

$\frac{d x^{2} y \left(x^{2} + 4 x + 6 y\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{d x^{2} y \left(x^{2} + 4 x + 6 y\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                 
 |                                                             / 4       \      2  2
 | //         2\       / 3        \    \               3       |x       2|   d*x *y 
 | \\3*x*y + y /*d*x + \x  + 2*x*y/*d*y/ dx = C + d*y*x  + d*y*|-- + y*x | + -------
 |                                                             \4        /      2   
/

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

Simplify

The answer [src]

     2                         2  2        4
3*d*y    5*d*y        3   3*d*l *y    d*y*l 
------ + ----- - d*y*l  - --------- - ------
  2        4                  2         4

$$- \frac{d l^{4} y}{4} - d l^{3} y - \frac{3 d l^{2} y^{2}}{2} + \frac{3 d y^{2}}{2} + \frac{5 d y}{4}$$

     2                         2  2        4
3*d*y    5*d*y        3   3*d*l *y    d*y*l 
------ + ----- - d*y*l  - --------- - ------
  2        4                  2         4

$$- \frac{d l^{4} y}{4} - d l^{3} y - \frac{3 d l^{2} y^{2}}{2} + \frac{3 d y^{2}}{2} + \frac{5 d y}{4}$$

3*d*y^2/2 + 5*d*y/4 - d*y*l^3 - 3*d*l^2*y^2/2 - d*y*l^4/4

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

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Identical expressions

Similar expressions

Integral of (3xy+y^2)dx+(x^3+2xy)dy dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (3*x*y+y^2)*d*x+(x^3+2*x*y)*d*y dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (3xy+y^2)dx+(x^3+2xy)dy dx