Solve ∫ (from a to x) of xftdt

$$\int_{ a }^{ x } xft d t$$

Evaluate

$$\frac{fx\left(x^{2}-a^{2}\right)}{2}$$

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.

$$\frac{1}{2}xfx^{2}-\frac{1}{2}xfa^{2}$$

Simplify.

$$\frac{xf\left(x-a\right)\left(x+a\right)}{2}$$

Differentiate w.r.t. x

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