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

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

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integral from 0 to x of tf(t)

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

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

Solution steps

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

Take the constant out:

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

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

Simplify

tt : t^{2}

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

Apply the Power Rule

= f [\frac{t ^{3}}{3}]_{0}^{x}

Compute the boundaries:

\frac{x ^{3}}{3}

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

Simplify

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

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