Chapter 5

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(x^{3}+\sqrt{x}-9\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(x^{2 / 3}-\frac{1}{x}+5+\sqrt{x}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(x^{4}-5 e^{-2 x}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(2 \sqrt[3]{s}+\frac{5}{s}\right) d s\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{5 x^{3}-3}{x}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{3 e^{-x}+2 e^{3 x}}{e^{2 x}}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(t^{5}-3 t^{2}+\frac{1}{t^{2}}\right) d t\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int(x+1)\left(2 x^{2}+\sqrt{x}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int \sqrt{3 x+1} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int(3 x+1) \sqrt{3 x^{2}+2 x+5} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int(x+2)\left(x^{2}+4 x+2\right)^{5} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int \frac{x+2}{x^{2}+4 x+2} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int \frac{3 x+6}{\left(2 x^{2}+8 x+3\right)^{2}} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int(t-5)^{12} d t\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int v(v-5)^{12} d v\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int \frac{\ln (3 x)}{x} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int 5 x e^{-x^{2}} d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{x}{x-4}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{\sqrt{\ln x}}{x}\right) d x\)

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{e^{x}}{e^{x}+5}\right) d x\)

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{0}^{1}\left(5 x^{4}-8 x^{3}+1\right) d x\)

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{1}^{4}\left(\sqrt{t}+t^{-3 / 2}\right) d t\)

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{1}^{9} \frac{x^{2}+\sqrt{x}-5}{x} d x\)

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{-1}^{2} 30(5 x-2)^{2} d x\)

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{0}^{e-1}\left(\frac{x}{x+1}\right) d x\)

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{e}^{e^{2}} \frac{1}{x(\ln x)^{2}} d x\)

AREA BETWEEN CURVES In Exercises 31 through 38 , sketch the indicated region \(R\) and find its area by integration. \(R\) is the region under the curve \(y=e^{x}+e^{-x}\) over the interval \(-1 \leq x \leq 1\).

AREA BETWEEN CURVES In Exercises 31 through 38 , sketch the indicated region \(R\) and find its area by integration. \(R\) is the region under the curve \(y=\sqrt{9-5 x^{2}}\) over the interval \(0 \leq x \leq 1\).

AREA BETWEEN CURVES In Exercises 31 through 38 , sketch the indicated region \(R\) and find its area by integration. \(R\) is the region bounded by the curve \(y=\frac{4}{x}\) and the line \(x+y=5\).

AREA BETWEEN CURVES In Exercises 31 through 38 , sketch the indicated region \(R\) and find its area by integration. \(R\) is the region bounded by the curves \(y=\frac{8}{x}\) and \(y=\sqrt{x}\) and the line \(x=8 .\)

AREA BETWEEN CURVES In Exercises 31 through 38 , sketch the indicated region \(R\) and find its area by integration. \(R\) is the region bounded by the curve \(y=2+x-x^{2}\) and the \(x\) axis.

AVERAGE VALUE OF A FUNCTION In Exercises 39 through 42, find the average value of the given function over the indicated interval. \(f(x)=x^{3}-3 x+\sqrt{2 x}\); over \(1 \leq x \leq 8\)

AVERAGE VALUE OF A FUNCTION In Exercises 39 through 42, find the average value of the given function over the indicated interval. \(f(t)=t \sqrt[3]{8-7 t^{2}} ;\) over \(0 \leq t \leq 1\)

AVERAGE VALUE OF A FUNCTION In Exercises 39 through 42, find the average value of the given function over the indicated interval. \(g(v)=v e^{-v^{2}} ;\) over \(0 \leq v \leq 2\)

AVERAGE VALUE OF A FUNCTION In Exercises 39 through 42, find the average value of the given function over the indicated interval. \(h(x)=\frac{e^{x}}{1+2 e^{x}} ;\) over \(0 \leq x \leq 1\)

CONSUMERS' SURPLUS In Exercises 43 through 46, \(p=D(q)\) is the demand curve for a particular commod- ity; that is, \(q\) units of the commodity will be demanded when the price is \(p=D(q)\) dollars per unit. In each case, for the given level of production \(q_{0}\), find \(p_{0}=D\left(q_{0}\right)\) and compute the corresponding consumers' surplus. \(D(q)=4\left(36-q^{2}\right) ; q_{0}=2\) units

CONSUMERS' SURPLUS In Exercises 43 through 46, \(p=D(q)\) is the demand curve for a particular commod- ity; that is, \(q\) units of the commodity will be demanded when the price is \(p=D(q)\) dollars per unit. In each case, for the given level of production \(q_{0}\), find \(p_{0}=D\left(q_{0}\right)\) and compute the corresponding consumers' surplus. . \(D(q)=100-4 q-3 q^{2} ; q_{0}=5\) units

CONSUMERS' SURPLUS In Exercises 43 through 46, \(p=D(q)\) is the demand curve for a particular commod- ity; that is, \(q\) units of the commodity will be demanded when the price is \(p=D(q)\) dollars per unit. In each case, for the given level of production \(q_{0}\), find \(p_{0}=D\left(q_{0}\right)\) and compute the corresponding consumers' surplus. \(D(q)=10 e^{-0.1 q} ; q_{0}=4\) units

CONSUMERS' SURPLUS In Exercises 43 through 46, \(p=D(q)\) is the demand curve for a particular commod- ity; that is, \(q\) units of the commodity will be demanded when the price is \(p=D(q)\) dollars per unit. In each case, for the given level of production \(q_{0}\), find \(p_{0}=D\left(q_{0}\right)\) and compute the corresponding consumers' surplus. \(D(q)=5+3 e^{-0.2 q} ; q_{0}=10\) units

LORENZ CURVES In Exerises 47 through 50 , sketch the Lorenz curve \(y=L(x)\) and find the corresponding Gini index. L(x)=x^{3 / 2}$

LORENZ CURVES In Exerises 47 through 50 , sketch the Lorenz curve \(y=L(x)\) and find the corresponding Gini index. \(L(x)=x^{1.2}\)

LORENZ CURVES In Exerises 47 through 50 , sketch the Lorenz curve \(y=L(x)\) and find the corresponding Gini index. \(L(x)=0.3 x^{2}+0.7 x\)

LORENZ CURVES In Exerises 47 through 50 , sketch the Lorenz curve \(y=L(x)\) and find the corresponding Gini index. \(L(x)=0.75 x^{2}+0.25 x\)

SURVIVAL AND RENEWAL In Exercises 51 through 54, an initial population \(P_{0}\) is given along with a renewal rate \(R(t)\) and a survival function \(S(t)\). In each case, use the given information to find the population at the end of the indicated term \(T\). \(P_{0}=75,000 ; R(t)=60 ; S(t)=e^{-0.09 t} ; t\) in months; term \(T=6\) months

SURVIVAL AND RENEWAL In Exercises 51 through 54, an initial population \(P_{0}\) is given along with a renewal rate \(R(t)\) and a survival function \(S(t)\). In each case, use the given information to find the population at the end of the indicated term \(T\). \(P_{0}=125,000 ; R(t)=250 ; S(t)=e^{-0.015 t} ; t\) in years; term \(T=5\) years

SURVIVAL AND RENEWAL In Exercises 51 through 54, an initial population \(P_{0}\) is given along with a renewal rate \(R(t)\) and a survival function \(S(t)\). In each case, use the given information to find the population at the end of the indicated term \(T\). \(P_{0}=100,000 ; R(t)=90 e^{0.1 t} ; S(t)=e^{-0.2 t} ; t\) in years; term \(T=10\) years

SURVIVAL AND RENEWAL In Exercises 51 through 54, an initial population \(P_{0}\) is given along with a renewal rate \(R(t)\) and a survival function \(S(t)\). In each case, use the given information to find the population at the end of the indicated term \(T\). \(P_{0}=200,000 ; R(t)=50 e^{0.12 t} ; S(t)=e^{-0.017 t} ; t\) in hours; term \(T=20\) hours

VOLUME OF SOLID OF REVOLUTION In Exercises 55 through 58 , find the volume of the solid of revolution formed by rotating the specified region \(R\) about the \(x\) axis. \(R\) is the region under the curve \(y=x^{2}+1\) from \(x=-1\) to \(x=2\).

VOLUME OF SOLID OF REVOLUTION In Exercises 55 through 58 , find the volume of the solid of revolution formed by rotating the specified region \(R\) about the \(x\) axis. \(R\) is the region under the curve \(y=e^{-x / 10}\) from \(x=0\) to \(x=10\).

VOLUME OF SOLID OF REVOLUTION In Exercises 55 through 58 , find the volume of the solid of revolution formed by rotating the specified region \(R\) about the \(x\) axis. \(R\) is the region under the curve \(y=\frac{1}{\sqrt{x}}\) from \(x=1\) to \(x=3\).