What is the integral from 0 to θ of x*(2x)/(θ^2) ?

The integral from 0 to θ of x*(2x)/(θ^2) is (2θ)/3

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integral from 0 to θ of x*(2x)/(θ^2)

\int_{0}^{θ} x \cdot \frac{2 x}{θ ^{2}} d x

\frac{2 θ}{3}

Solution steps

\int_{0}^{θ} x \frac{2 x}{θ ^{2}} d x

Take the constant out:

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

= \frac{2}{θ ^{2}} \cdot \int_{0}^{θ} xx d x

Simplify

xx : x^{2}

= \frac{2}{θ ^{2}} \cdot \int_{0}^{θ} x^{2} d x

Apply the Power Rule

: [\frac{x ^{3}}{3}]_{0}^{θ}

= \frac{2}{θ ^{2}} [\frac{x ^{3}}{3}]_{0}^{θ}

Compute the boundaries:

\frac{θ ^{3}}{3}

= \frac{2}{θ ^{2}} \cdot \frac{θ ^{3}}{3}

Simplify

= \frac{2 θ}{3}

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What is the integral from 0 to θ of x*(2x)/(θ^2) ?
The integral from 0 to θ of x*(2x)/(θ^2) is (2θ)/3