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Identical expressions
(x^ two +3x*y)*d*y
(x squared plus 3x multiply by y) multiply by d multiply by y
(x to the power of two plus 3x multiply by y) multiply by d multiply by y
(x2+3x*y)*d*y
x2+3x*y*d*y
(x²+3x*y)*d*y
(x to the power of 2+3x*y)*d*y
(x^2+3xy)dy
(x2+3xy)dy
x2+3xydy
x^2+3xydy
(x^2+3x*y)*d*ydx
Similar expressions
(x^2-3x*y)*d*y

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

The solution

You have entered [src]

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

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

$\frac{d x^{2} y \left(2 x + 9 y\right)}{6}$
Add the constant of integration:

$\frac{d x^{2} y \left(2 x + 9 y\right)}{6}+ \mathrm{constant}$

The answer is:

$\frac{d x^{2} y \left(2 x + 9 y\right)}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$d\,y\,\left({{3\,x^2\,y}\over{2}}+{{x^3}\over{3}}\right)$$

Simplify

The answer [src]

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

$$d\,y\,\left({{9\,y+2}\over{6}}-{{9\,d^2\,y+2\,d^3}\over{6}}\right)$$

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

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

Simplify

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

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