91Ó°ÊÓ

Establish the following reduction formula: $$ \int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x $$

The marginal cost, in thousands of dollars, of a brick manufacturer is given by \(\frac{x}{\sqrt{x^{2}+9}},\) where \(x\) is in thousands of bricks. If fixed costs are \(\$ 10,000,\) find \(C=C(x)\).

By making the change of variable \(x=-\ln u,\) show that the improper integral \(\int_{0}^{\infty} f(x) d x\) becomes \(\int_{0}^{1} u^{-1} f(-\ln u) d u .\) Use this and Simpson's rule for \(n=128\) to approximate \(\int_{0}^{\infty} \frac{e^{-x}}{1+x^{2}} d x\).

If \(f(0)=g(0)=0,\) show that $$ \begin{aligned} \int_{0}^{a} f(x) g^{\prime \prime}(x) d x=& f(a) g^{\prime}(a)-f^{\prime}(a) g(a) \\ &+\int_{0}^{a} f^{\prime \prime}(x) g(x) d x \end{aligned} $$

Use integration by parts to show that $$ \int f(x) d x=x f(x)-\int x f^{\prime}(x) d x $$

Find the consumers' surplus if $$p=D(x)=\frac{4 e}{e+e^{0.10 x}}$$ is the demand equation with \(x\) measured in thousands of units and \(p_{0}=2\) is the equilibrium price of the commodity.

\(f(t)\) is the rate of change of total income per year, and \(r\) is the annual interest rate compounding continuously. Find \(P_{V}(\infty)\). $$ f(t)=100, r=8 \% $$

\(f(t)\) is the rate of change of total income per year, and \(r\) is the annual interest rate compounding continuously. Find \(P_{V}(\infty)\). $$ f(t)=1000, r=12 \% $$

\(f(t)\) is the rate of change of total income per year, and \(r\) is the annual interest rate compounding continuously. Find \(P_{V}(\infty)\). $$ f(t)=100 e^{-0.1 t}, r=8 \% $$

The concentration (density) of pollutants, measured in thousands of particles per mile per day, at a distance of \(x\) miles east of an industrial plant is given by $$\delta(x)=\frac{1}{x^{2}+5 x+4}$$ Find the amount of pollutants between \(x=0\) and \(x=5\) miles.