Solve ∫ (from 2 to 3) of x_{0}+v_{0}t+1/2at^2+253972

$$\int _ { 2 } ^ { 3 } x _ { 0 } + v _ { 0 } t + \frac { 1 } { 2 } a t ^ { 2 } + 253972$$

Evaluate

$$\frac{at^{2}}{2}+tv_{0}+x_{0}+253972$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x_{0}+v_{0}t+\frac{at^{2}}{2}+253972\mathrm{d}x$$

Find the integral of x_{0}+v_{0}t+\frac{at^{2}}{2}+253972 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(x_{0}+v_{0}t+\frac{at^{2}}{2}+253972\right)x$$

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.

$$\left(x_{0}+v_{0}t+\frac{1}{2}at^{2}+253972\right)\times 3-\left(x_{0}+v_{0}t+\frac{1}{2}at^{2}+253972\right)\times 2$$

Simplify.

$$x_{0}+\frac{at^{2}}{2}+tv_{0}+253972$$

Differentiate w.r.t. x_0

$$1$$