What is the integral from 0 to z of 6xz(x+z) ?

The integral from 0 to z of 6xz(x+z) is 5z^4

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integral from 0 to z of 6xz(x+z)

\int_{0}^{z} 6 x z (x + z) d x

5 z^{4}

Solution steps

\int_{0}^{z} 6 x z (x + z) d x

Take the constant out:

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

= 6 z \cdot \int_{0}^{z} (z + x) x d x

Expand

(z + x) x : z x + x^{2}

= 6 z \cdot \int_{0}^{z} z x + x^{2} 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

= 6 z (\int_{0}^{z} z x d x + \int_{0}^{z} x^{2} d x)

\int_{0}^{z} z x d x = \frac{z ^{3}}{2}

\int_{0}^{z} x^{2} d x = \frac{z ^{3}}{3}

= 6 z (\frac{z ^{3}}{2} + \frac{z ^{3}}{3})

Simplify

6 z (\frac{z ^{3}}{2} + \frac{z ^{3}}{3}) : 5 z^{4}

= 5 z^{4}

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What is the integral from 0 to z of 6xz(x+z) ?
The integral from 0 to z of 6xz(x+z) is 5z^4