Chapter 11

Use a symbolic integration utility to find the indefinite integral. $$ \int(x+1)(3 x-2) d x $$

Find the consumer and producer surpluses. $$ p_{1}(x)=300-x \quad p_{2}(x)=100+x $$

Evaluate the definite integral. $$ \int_{-1}^{1}\left(e^{x}-e^{-x}\right) d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{2\left(e^{x}-e^{-x}\right)}{\left(e^{x}+e^{-x}\right)^{2}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int \sqrt{x}\left(4-x^{3 / 2}\right)^{2} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(2 t^{2}-1\right)^{2} d t $$

Evaluate the definite integral. $$ \int_{0}^{1} e^{2 x} \sqrt{e^{2 x}+1} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{1}{(x-1)^{2}} d x $$

(a) perform the integration in two ways: once using the Simple Power Rule and once using the General Power Rule. (b) Explain the difference in the results. (c) Which method do you prefer? Explain your reasoning. $$ \int(x-1)^{2} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int y^{2} \sqrt{y} d y $$

Evaluate the definite integral. $$ \int_{0}^{1} \frac{e^{-x}}{\sqrt{e^{-x}+1}} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{1}{\sqrt{x+1}} d x $$

(a) perform the integration in two ways: once using the Simple Power Rule and once using the General Power Rule. (b) Explain the difference in the results. (c) Which method do you prefer? Explain your reasoning. $$ \int(3-x)^{2} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int(1+3 t) t^{2} d t $$

Evaluate the definite integral. $$ \int_{0}^{2} \frac{x}{1+4 x^{2}} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int 4 e^{2 x-1} d x $$

(a) perform the integration in two ways: once using the Simple Power Rule and once using the General Power Rule. (b) Explain the difference in the results. (c) Which method do you prefer? Explain your reasoning. $$ \int x\left(x^{2}-1\right)^{2} d x $$

Evaluate the definite integral. $$ \int_{0}^{1} \frac{e^{2 x}}{e^{2 x}+1} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int\left(5 e^{-2 x}+1\right) d x $$

(a) perform the integration in two ways: once using the Simple Power Rule and once using the General Power Rule. (b) Explain the difference in the results. (c) Which method do you prefer? Explain your reasoning. $$ \int x\left(2 x^{2}+1\right)^{2} d x $$

Two models, \(R_{1}\) and \(R_{2}\), are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with \(t=7\) corresponding to \(2007 .\) Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period? $$ R_{1}=7.21+0.58 t, R_{2}=7.21+0.45 t $$

Evaluate the definite integral by the most convenient method. Explain your approach. $$ \int_{-1}^{1}|4 x| d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{x^{3}-8 x}{2 x^{2}} d x $$

Two models, \(R_{1}\) and \(R_{2}\), are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with \(t=7\) corresponding to \(2007 .\) Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period? $$ R_{1}=7.21+0.26 t+0.02 t^{2}, R_{2}=7.21+0.1 t+0.01 t^{2} $$

Evaluate the definite integral by the most convenient method. Explain your approach. $$ \int_{0}^{3}|2 x-3| d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{x-1}{4 x} d x $$

The projected fuel cost \(C\) (in millions of dollars per year) for an airline company from 2007 through 2013 is \(C_{1}=568.5+7.15 t\), where \(t=7\) corresponds to \(2007 .\) If the company purchases more efficient airplane engines, fuel cost is expected to decrease and to follow the model \(C_{2}=525.6+6.43 t\). How much can the company save with the more efficient engines? Explain your reasoning.

Evaluate the definite integral by the most convenient method. Explain your approach. $$ \int_{0}^{4}(2-|x-2|) d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{2}{1+e^{-x}} d x $$

Cost The marginal cost of a product is modeled by \(\frac{d C}{d x}=\frac{4}{\sqrt{x+1}} .\) When \(x=15, C=50\) (a) Find the cost function. (b) Use a graphing utility to graph \(d C / d x\) and \(C\) in the same viewing window.

Find the particular solution \(y=f(x)\) that satisfies the differential equation and initial condition. $$ f^{\prime}(x)=4 x ; \quad f(0)=6 $$

Health An epidemic was spreading such that \(t\) weeks after its outbreak it had infected \(N_{1}(t)=0.1 t^{2}+0.5 t+150, \quad 0 \leq t \leq 50\) people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the number of people infected was governed by the model \(N_{2}(t)=-0.2 t^{2}+6 t+200\)

Evaluate the definite integral by the most convenient method. Explain your approach. $$ \int_{-4}^{4}(4-|x|) d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{3}{1+e^{-3 x}} d x $$

Cost The marginal cost of a product is modeled by \(\frac{d C}{d x}=\frac{12}{\sqrt[3]{12 x+1}}\) When \(x=13, C=100\). (a) Find the cost function. (b) Use a graphing utility to graph \(d C / d x\) and \(C\) in the same viewing window.

Find the particular solution \(y=f(x)\) that satisfies the differential equation and initial condition. $$ f^{\prime}(x)=\frac{1}{5} x-2 ; \quad f(10)=-10 $$

Consumer Trends For the years 1996 through 2004 , the per capita consumption of fresh pineapples (in pounds per year) in the United States can be modeled by \(C(t)=\left\\{\begin{array}{c}-0.046 t^{2}+1.07 t-2.9,6 \leq t \leq 10 \\\ -0.164 t^{2}+4.53 t-26.8,10

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. $$ \int_{-1}^{2} \frac{x}{x^{2}-9} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{x^{2}+2 x+5}{x-1} d x $$

Find the supply function \(x=f(p)\) that satisfies the initial conditions. $$ \frac{d x}{d p}=p \sqrt{p^{2}-25}, \quad x=600 \text { when } p=\$ 13 $$

Find the particular solution \(y=f(x)\) that satisfies the differential equation and initial condition. $$ f^{\prime}(x)=2(x-1) ; \quad f(3)=2 $$

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. $$ \int_{2}^{3} \frac{x+1}{x^{2}+2 x-3} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{x-3}{x+3} d x $$

Find the supply function \(x=f(p)\) that satisfies the initial conditions. $$ \frac{d x}{d p}=\frac{10}{\sqrt{p-3}}, \quad x=100 \text { when } p=\$ 3 $$

Find the particular solution \(y=f(x)\) that satisfies the differential equation and initial condition. $$ f^{\prime}(x)=(2 x-3)(2 x+3) ; \quad f(3)=0 $$

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. $$ \begin{aligned} &\int_{0}^{3} \frac{2 e^{x}}{2+e^{x}} d x\\\ &\text { Exercises } 55-60, \text { eva } \end{aligned} $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{1+e^{-x}}{1+x e^{-x}} d x $$

Find the demand function \(x=f(p)\) that satisfies the initial conditions. $$ \frac{d x}{d p}=-\frac{6000 p}{\left(p^{2}-16\right)^{3 / 2}}, \quad x=5000 \text { when } p=\$ 5 $$

Find the particular solution \(y=f(x)\) that satisfies the differential equation and initial condition. $$ f^{\prime}(x)=\frac{2-x}{x^{3}}, x>0 ; \quad f(2)=\frac{3}{4} $$

The revenue from a manufacturing process (in millions of dollars per year) is projected to follow the model \(R=100\) for 10 years. Over the same period of time, the cost (in millions of dollars per year) is projected to follow the model \(C=60+0.2 t^{2}\), where \(t\) is the time (in years). Approximate the profit over the 10 -year period.