Solve ∫ (from - 3 to 1) of x(t)dt

$$\int _ { - 3 } ^ { 1 } x ( t ) d t$$

Evaluate

$$-4x$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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}.

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

Simplify.

$$\frac{xt^{2}}{2}$$

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.

$$\frac{1}{2}x\times 1^{2}-\frac{1}{2}x\left(-3\right)^{2}$$

Simplify.

$$-4x$$

Differentiate w.r.t. x

$$-4$$