What is the integral from 0 to y of 2(x+y) ?

The integral from 0 to y of 2(x+y) is 3y^2

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integral from 0 to y of 2(x+y)

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

3 y^{2}

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{y} y + x d x

Apply the Sum Rule:

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

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

\int_{0}^{y} y d x = y^{2}

\int_{0}^{y} x d x = \frac{y ^{2}}{2}

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

Simplify

2 (y^{2} + \frac{y ^{2}}{2}) : 3 y^{2}

= 3 y^{2}

What is the integral from 0 to y of 2(x+y) ?
The integral from 0 to y of 2(x+y) is 3y^2