Solve ∫ (from 0 to x) of xf(t)dt)^prime

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

Evaluate

$$\frac{fx^{3}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int xft\mathrm{d}t$$

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

$$xf\int t\mathrm{d}t$$

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

$$xf\times \frac{t^{2}}{2}$$

Simplify.

$$\frac{xft^{2}}{2}$$

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.

$$\frac{1}{2}xfx^{2}-\frac{1}{2}xf\times 0^{2}$$

Simplify.

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

Differentiate w.r.t. x

$$\frac{3fx^{2}}{2}$$