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 (\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}

\int_{- 1}^{1} - x^{2} + 2 d x

\int_{- 4}^{2} \frac{2 x}{3 x ^{2} + 10} d x

\int_{0}^{3} \frac{9 - x ^{2}}{3} d x

\int_{1}^{e^{9}} \frac{10 ln ( x )}{x} d x

\int_{4}^{8} x^{3} d x

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