What is the integral from 0 to t of x^2g(x) ?

The integral from 0 to t of x^2g(x) is (gt^4)/4

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integral from 0 to t of x^2g(x)

\int_{0}^{t} x^{2} g (x) d x

\frac{g t ^{4}}{4}

Solution steps

\int_{0}^{t} x^{2} gx d x

Take the constant out:

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

= g \cdot \int_{0}^{t} x x^{2} d x

Simplify

x x^{2} : x^{3}

= g \cdot \int_{0}^{t} x^{3} d x

Apply the Power Rule

: [\frac{x ^{4}}{4}]_{0}^{t}

= g [\frac{x ^{4}}{4}]_{0}^{t}

Compute the boundaries:

\frac{t ^{4}}{4}

= g \frac{t ^{4}}{4}

Simplify

= \frac{g t ^{4}}{4}

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What is the integral from 0 to t of x^2g(x) ?
The integral from 0 to t of x^2g(x) is (gt^4)/4