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Solving integrals/ (x^2-2xy)+(2xy+y^2)

Integral of (x^2-2xy)+(2xy+y^2) dx

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The solution

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∫01((x2−2xy)+(2xy+y2)) dx\int\limits_{0}^{1} \left(\left(x^{2} - 2 x y\right) + \left(2 x y + y^{2}\right)\right)\, dx 
Integral(x^2 - 2*x*y + (2*x)*y + y^2, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    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:

        ∫(−2xy) dx=−y∫2x dx\int \left(- 2 x y\right)\, dx = - y \int 2 x\, dx

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

          ∫2x dx=2∫x dx\int 2 x\, dx = 2 \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: x2x^{2}

        So, the result is: −x2y- x^{2} y

      The result is: x33−x2y\frac{x^{3}}{3} - x^{2} y

    1. Integrate term-by-term:

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

        ∫2xy dx=y∫2x dx\int 2 x y\, dx = y \int 2 x\, dx

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

          ∫2x dx=2∫x dx\int 2 x\, dx = 2 \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: x2x^{2}

        So, the result is: x2yx^{2} y

      1. The integral of a constant is the constant times the variable of integration:

        ∫y2 dx=xy2\int y^{2}\, dx = x y^{2}

      The result is: x2y+xy2x^{2} y + x y^{2}

    The result is: x33+xy2\frac{x^{3}}{3} + x y^{2}

  2. Now simplify:

    x(x23+y2)x \left(\frac{x^{2}}{3} + y^{2}\right)

  3. Add the constant of integration:

    x(x23+y2)+constantx \left(\frac{x^{2}}{3} + y^{2}\right)+ \mathrm{constant}

The answer is:

x(x23+y2)+constantx \left(\frac{x^{2}}{3} + y^{2}\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | \x  - 2*x*y + 2*x*y + y / dx = C + -- + x*y 
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∫((x2−2xy)+(2xy+y2)) dx=C+x33+xy2\int \left(\left(x^{2} - 2 x y\right) + \left(2 x y + y^{2}\right)\right)\, dx = C + \frac{x^{3}}{3} + x y^{2}
Simplify
The answer [src] 
1    2
- + y 
3
y2+13y^{2} + \frac{1}{3} 
=
= 
1    2
- + y 
3
y2+13y^{2} + \frac{1}{3} 
1/3 + y^2

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

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