Solve ∫ (from 0 to x) of tf(t)dt

$$\int _ { 0 } ^ { x } t f ( t ) d t$$

Evaluate

$$\frac{fx^{3}}{3}$$

Solution Steps

Multiply t and t to get t^{2}.

$$\int _{0}^{x}t^{2}f\mathrm{d}t$$

Evaluate the indefinite integral first.

$$\int t^{2}f\mathrm{d}t$$

Factor out the constant using \int af\left(t\right)\mathrm{d}t=a\int f\left(t\right)\mathrm{d}t.

$$f\int t^{2}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}.

$$f\times \frac{t^{3}}{3}$$

Simplify.

$$\frac{ft^{3}}{3}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{3}fx^{3}-\frac{1}{3}f\times 0^{3}$$

Simplify.

$$\frac{fx^{3}}{3}$$

Differentiate w.r.t. x

$$fx^{2}$$