In the manufacture of a product, the marginal cost of producing $x$ units is $C^{\prime}(x)$ and fixed costs are $C(0)$. Find the total cost function $C(x)$ when:

(a) $C^{\prime}(x)=3 x+4, C(0)=40$

(b) $C^{\prime}(x)=a x+b, C(0)=C_0$

Find the following integrals:

(a) $\int_0^1 x \sqrt{1+x^2} d x$

(b) $\int_1^e \frac{\ln y}{y} d y$

(c) $\int_1^3 \frac{1}{x^2} e^{2 / x} d x$

(d) $\int_5^8 \frac{x}{x-4} d x$

(In (d) you should try at least two different methods.)

In connection with Example 1, find the following:

(a) $\int_0^{\infty} x \lambda e^{-\lambda x} d x$

(b) $\int_0^{\infty}(x-1 / \lambda)^2 \lambda e^{-\lambda x} d x$

(c) $\int_0^{\infty}(x-1 / \lambda)^3 \lambda e^{-\lambda x} d x$

(The three numbers you obtain are called respectively the expectation, the variance, and the third central moment of the exponential distribution.)

Evaluate the following integrals:

(a) $\int_0^{12} 50 d x$

(b) $\int_0^2\left(x-\frac{1}{2} x^2\right) d x$

(c) $\int_{-3}^3(u+1)^2 d u$

(d) $\int_1^5 \frac{2}{z} d z$

(e) $\int_2^{12} \frac{3 d t}{t+4} d t$

(f) $\int_0^4 v \sqrt{v^2+9} d v$

Suppose that $y=\alpha k e^{\beta t}$ denotes production as a function of capital $k$, where the factor $e^{\beta t}$ is due to technical progress. Suppose that a constant fraction $s \in(0,1)$ is saved, and that capital accumulation is equal to savings, so that we have the separable differential equation

$$\dot{k}=s \alpha k e^{\beta t}, \quad k(0)=k_0$$

The constants $\alpha, \beta$, and $k_0$ are positive. Find the solution.

Let $N(t)$ denote the number of people in a country whose homes have broadband internet. Suppose that the rate at which new people get access is proportional to the number of people who still have no access. If the population size is $P$, the differential equation for $N(t)$ is then

$$\dot{N}(t)=k(P-N(t))$$

where $k$ is a positive constant. Find the solution of this equation if $N(0)=0$. Then find the limit of $N(t)$ as $t \rightarrow \infty$.

Compute the area $A$ bounded by the graph of $f(x)=1 / x^3$, the $x$-axis, and the two lines $x=-2$ and $x=-1$. Make a drawing. (Hint: $f(x)<0$ in $[-2,-1]$.)

Suppose that a firm's capital stock $K(t)$ satisfies the differential equation

$$\dot{K}(t)=I-\delta K(t)$$

where investment $I$ is constant, and $\delta K(t)$ denotes depreciation, with $\delta$ a positive constant.

(a) Find the solution of the equation if the capital stock at time $t=0$ is $K_0$.

(b) Let $\delta=0.05$ and $I=10$. Explain what happens as $t \rightarrow \infty$ when:

(i) $K_0=150$;

(ii) $K_0=250$.

Define $f$ for all $x$ by $f(x)=1 /(b-a)$ for $x \in[a, b]$, and $f(x)=0$ for $x \notin[a, b]$, where $b>a$. (In statistics, $f$ is the density function of the uniform (or rectangular) distribution on the interval $[a, b]$.) Find the following:

(a) $\int_{-\infty}^{+\infty} f(x) d x$

(b) $\int_{-\infty}^{+\infty} x f(x) d x$

(c) $\int_{-\infty}^{+\infty} x^2 f(x) d x$

(a) Follow the pattern in Example 1 and find the mean income $m$ over the interval $[b, 2 b]$ when $f(r)=B r^{-2}$, assuming that there are $n$ individuals in the population.

(b) Assume that the individuals' demand function is $D(p, r)=A p^\gamma r^\delta$ with $A>0 \gamma<0$, $\delta>0, \delta \neq 1$. Compute the total demand $x(p)$ by using formula (7).

Use integration by parts to evaluate the following:

(a) $\int_{-1}^1 x \ln (x+2) d x$

(b) $\int_0^2 x 2^x d x$

(c) $\int_0^1 x^2 e^x d x$

(d) $\int_0^3 x \sqrt{1+x} d x$

(In (d) you should graph the integrand and decide if your answer is reasonable.)

If $\int_a^b f(x) d x=8$ and $\int_a^c f(x) d x=4$, what is $\int_c^b f(x) d x$ ?

Find the following integrals by means of an appropriate substitution:

(a) $\int x\left(2 x^2+3\right)^5 d x$

(b) $\int x^2 e^{x^3+2} d x$

(c) $\int \frac{\ln (x+2)}{2 x+4} d x$

(d) $\int x \sqrt{1+x} d x$

(e) $\int \frac{x^3}{\left(1+x^2\right)^3} d x$

(f) $\int x^5 \sqrt{4-x^3} d x$

Find the following integrals:

(a) $\int e^{-x} d x$

(b) $\int e^{\frac{1}{4} x} d x$

(c) $\int 3 e^{-2 x} d x$

(d) $\int 2^x d x$