Find the family of all antiderivatives of each derivative.

$$\frac{dm}{dn}=10n(n^{2}-8)^{7}$$

Find the derivative or indefinite integral as indicated.

$$\frac{d}{dt}\bigg(\displaystyle{\int}\frac{\ln{t}}{t}\ dt\bigg)$$

Find the derivative or indefinite integral as indicated.

$$\frac{d}{dx}\bigg(\displaystyle{\int}x^{3}\ dx\bigg)$$

Salvage value. A new piece of industrial equipment will depreciate in value, rapidly at first and then less rapidly as time goes on. Suppose that the rate (in dollars per year) at which the book value of a new milling machine changes is given approximately by

$$V'(t)=f(t)=500(t-12)\qquad 0\leq{t}\leq10$$

where $V(t)$ is the value of the machine after $t$ years. What is the total loss in value of the machine in the first $5$ years? In the second $5$ years? Set up appropriate integrals and solve.

The left sum $L_{n}$ or the right sum $R_{n}$ is used to approximate the definite integral to the indicated accuracy. How large must $n$ be chosen in each case? (Each function is increasing over the indicated interval.)

$$\displaystyle\int_{1}^{3}x^{x}\ dx=L_{n}\pm0.5$$