What is the integral from 0 to x+y of (x+y+z) ?

The integral from 0 to x+y of (x+y+z) is x^2+2xy+y^2+((x+y)^2)/2

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integral from 0 to x+y of (x+y+z)

\int_{0}^{x + y} (x + y + z) d z

x^{2} + 2 x y + y^{2} + \frac{( x + y ) ^{2}}{2}

Solution steps

\int_{0}^{x + y} x + y + z d z

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{x + y} x d z + \int_{0}^{x + y} y d z + \int_{0}^{x + y} z d z

\int_{0}^{x + y} x d z = x (x + y)

\int_{0}^{x + y} y d z = y (x + y)

\int_{0}^{x + y} z d z = \frac{( x + y ) ^{2}}{2}

= x (x + y) + y (x + y) + \frac{( x + y ) ^{2}}{2}

Simplify

x (x + y) + y (x + y) + \frac{( x + y ) ^{2}}{2} : x^{2} + 2 x y + y^{2} + \frac{( x + y ) ^{2}}{2}

= x^{2} + 2 x y + y^{2} + \frac{( x + y ) ^{2}}{2}

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What is the integral from 0 to x+y of (x+y+z) ?
The integral from 0 to x+y of (x+y+z) is x^2+2xy+y^2+((x+y)^2)/2