91Ó°ÊÓ

Evaluate. (Be sure to check by differentiating!) $$ \int\left(8+x^{3}\right)^{5} 3 x^{2} d x $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int 4 x e^{4 x} d x $$

Find each antiderivative using Table 1. $$ \int x e^{-3 x} d x $$

Find the area under the graph of \(f\) over [1,5]. $$ f(x)=\left\\{\begin{array}{lll} 2 x+1, & \text { for } & x \leq 3 \\ 10-x, & \text { for } & x>3 \end{array}\right. $$

Find the area under the given curve over the indicated interval. $$ y=4 ; \quad[1,3] $$

Calculate total cost (disregarding any fixed costs) or total profit. Total profit from marginal profit. A concert promoter sells \(x\) tickets and has a marginal-profit function given by $$P^{\prime}(x)=2 x-150$$ where \(P^{\prime}(x)\) is in dollars per ticket. This means that the rate of change of total profit with respect to the number of tickets sold, \(x\), is \(P^{\prime}(x)\). Find the total profit from the sale of the first 300 tickets.

Find each integral. $$ \int x^{6} d x $$

Find the area under the graph of \(f\) over [1,5]. $$ f(x)=\left\\{\begin{array}{ll} x+5, & \text { for } \quad x \leq 4 \\ 11-\frac{1}{2} x, & \text { for } \quad x>4 \end{array}\right. $$

Calculate total cost (disregarding any fixed costs) or total profit. Poyse Inc. has a marginal-profit function given by $$P^{\prime}(x)=-2 x+80$$ where \(P^{\prime}(x)\) is in dollars per unit. This means that the rate of change of total profit with respect to the number of units produced, \(x\), is \(P^{\prime}(x)\). Find the total profit from the production and sale of the first 40 units.

Find the area under the given curve over the indicated interval. $$ y=5 ; \quad[1,3] $$