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

The integral from 0 to x of 2/3 (x+2y) is (4x^2)/3

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integral from 0 to x of 2/3 (x+2y)

\int_{0}^{x} \frac{2}{3} (x + 2 y) d y

\frac{4 x ^{2}}{3}

Solution steps

\int_{0}^{x} \frac{2}{3} (x + 2 y) d y

Take the constant out:

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

= \frac{2}{3} \cdot \int_{0}^{x} x + 2 y d y

Apply the Sum Rule:

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

= \frac{2}{3} (\int_{0}^{x} x d y + \int_{0}^{x} 2 y d y)

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

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

= \frac{2}{3} (x^{2} + x^{2})

Simplify

\frac{2}{3} (x^{2} + x^{2}) : \frac{4 x ^{2}}{3}

= \frac{4 x ^{2}}{3}

What is the integral from 0 to x of 2/3 (x+2y) ?
The integral from 0 to x of 2/3 (x+2y) is (4x^2)/3