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

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d1dy(x2+3xy)dx\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
  1. The integral of a constant times a function is the constant times the integral of the function:

    dy(x2+3xy)dx=dy(x2+3xy)dx\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 xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

        x2dx=x33\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:

        3xydx=3yxdx\int 3 x y\, dx = 3 y \int x\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          xdx=x22\int x\, dx = \frac{x^{2}}{2}

        So, the result is: 3x2y2\frac{3 x^{2} y}{2}

      The result is: x33+3x2y2\frac{x^{3}}{3} + \frac{3 x^{2} y}{2}

    So, the result is: dy(x33+3x2y2)d y \left(\frac{x^{3}}{3} + \frac{3 x^{2} y}{2}\right)

  2. Now simplify:

    dx2y(2x+9y)6\frac{d x^{2} y \left(2 x + 9 y\right)}{6}

  3. Add the constant of integration:

    dx2y(2x+9y)6+constant\frac{d x^{2} y \left(2 x + 9 y\right)}{6}+ \mathrm{constant}


The answer is:

dx2y(2x+9y)6+constant\frac{d x^{2} y \left(2 x + 9 y\right)}{6}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                           
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 | \x  + 3*x*y/*d*y dx = C + d*y*|-- + ------|
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dy(3x2y2+x33)d\,y\,\left({{3\,x^2\,y}\over{2}}+{{x^3}\over{3}}\right)
The answer [src]
     3  2      4              2
  3*d *y    y*d    d*y   3*d*y 
- ------- - ---- + --- + ------
     2       3      3      2   
dy(9y+269d2y+2d36)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   
d4y33d3y22+3dy22+dy3- \frac{d^{4} y}{3} - \frac{3 d^{3} y^{2}}{2} + \frac{3 d y^{2}}{2} + \frac{d y}{3}

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