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

Integral of (x^2+y)+(2x-y) dy

Limits of integration:

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Piecewise:

The solution

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 |  \x  + y + 2*x - y/ dy
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a*b
∫ab1((2x−y)+(x2+y)) dy\int\limits_{a b}^{1} \left(\left(2 x - y\right) + \left(x^{2} + y\right)\right)\, dy 
Integral(x^2 + y + 2*x - y, (y, a*b, 1))
Detail solution

  1. Integrate term-by-term:

    1. Integrate term-by-term:

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

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

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

        ∫(−y) dy=−∫y dy\int \left(- y\right)\, dy = - \int y\, dy

        1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫y dy=y22\int y\, dy = \frac{y^{2}}{2}

        So, the result is: −y22- \frac{y^{2}}{2}

      The result is: 2xy−y222 x y - \frac{y^{2}}{2}

    1. Integrate term-by-term:

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

        ∫x2 dy=x2y\int x^{2}\, dy = x^{2} y

      1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫y dy=y22\int y\, dy = \frac{y^{2}}{2}

      The result is: x2y+y22x^{2} y + \frac{y^{2}}{2}

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

  2. Now simplify:

    xy(x+2)x y \left(x + 2\right)

  3. Add the constant of integration:

    xy(x+2)+constantx y \left(x + 2\right)+ \mathrm{constant}

The answer is:

xy(x+2)+constantx y \left(x + 2\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                        
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 | / 2              \             2        
 | \x  + y + 2*x - y/ dy = C + y*x  + 2*x*y
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∫((2x−y)+(x2+y)) dy=C+x2y+2xy\int \left(\left(2 x - y\right) + \left(x^{2} + y\right)\right)\, dy = C + x^{2} y + 2 x y
Simplify
The answer [src] 
 2             / 2      \
x  + 2*x - a*b*\x  + 2*x/
−ab(x2+2x)+x2+2x- a b \left(x^{2} + 2 x\right) + x^{2} + 2 x 
=
= 
 2             / 2      \
x  + 2*x - a*b*\x  + 2*x/
−ab(x2+2x)+x2+2x- a b \left(x^{2} + 2 x\right) + x^{2} + 2 x 
x^2 + 2*x - a*b*(x^2 + 2*x)

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

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