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

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

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The solution

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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 n≠−1n \neq -1:

        ∫x2 dx=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:

        ∫3xy dx=3y∫x dx\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 n≠−1n \neq -1:

          ∫x dx=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] 
  /                                           
 |                               / 3        2\
 | / 2        \                  |x    3*y*x |
 | \x  + 3*x*y/*d*y dx = C + d*y*|-- + ------|
 |                               \3      2   /
/
d y (3 x2 y2+x33)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 (9 y+26−9 d2 y+2 d36)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
−d4y3−3d3y22+3dy22+dy3- \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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