What is the integral from 0 to x of xf(t) ?

The integral from 0 to x of xf(t) is (x^3f)/2

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integral from 0 to x of xf(t)

\int_{0}^{x} x f (t) d t

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

Solution steps

\int_{0}^{x} x f t d t

Take the constant out:

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

= x f \cdot \int_{0}^{x} t d t

Apply the Power Rule

: [\frac{t ^{2}}{2}]_{0}^{x}

= x f [\frac{t ^{2}}{2}]_{0}^{x}

Compute the boundaries:

\frac{x ^{2}}{2}

= x f \frac{x ^{2}}{2}

Simplify

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

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What is the integral from 0 to x of xf(t) ?
The integral from 0 to x of xf(t) is (x^3f)/2