Integrals, Exponential Functions, and Logarithms

1. [latex]\frac{1}{x}[/latex]
2. [latex]-\frac{1}{x{(\text{ln}x)}^{2}}[/latex]
3. [latex]\text{ln}(x+1)+C[/latex]
4. [latex]\text{ln}(x)+1[/latex]
5. [latex]\cot (x)[/latex]
6. [latex]\frac{7}{x}[/latex]
7. [latex]\csc (x) \sec x[/latex]
8. [latex]-2 \tan x[/latex]
9. [latex]\frac{1}{2}\text{ln}(\frac{5}{3})[/latex]
10. [latex]2-\frac{1}{2}\text{ln}(5)[/latex]
11. [latex]\frac{1}{\text{ln}(2)}-1[/latex]
12. [latex]\frac{1}{2}\text{ln}(2)[/latex]
13. [latex]\frac{1}{3}{(\text{ln}x)}^{3}[/latex]
14. [latex]\frac{2{x}^{3}}{\sqrt{{x}^{2}+1}\sqrt{{x}^{2}-1}}[/latex]
15. [latex]{x}^{-2-(1\text{/}x)}(\text{ln}x-1)[/latex]
16. [latex]e{x}^{e-1}[/latex]
17. [latex]1[/latex]
18. [latex]-\frac{1}{{x}^{2}}[/latex]
19. [latex]\pi -\text{ln}(2)[/latex]
20. [latex]\frac{1}{x}[/latex]
21. [latex]{e}^{5}-6{\text{units}}^{2}[/latex]
22. [latex]\text{ln}(4)-1{\text{units}}^{2}[/latex]
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Exponential Growth and Decay

1. True
2. False; [latex]k=\frac{\text{ln}(2)}{t}[/latex]
3. [latex]20[/latex] hours
4. No. The relic is approximately [latex]871[/latex] years old.
5. [latex]71.92[/latex] years
6. [latex]5[/latex] days [latex]6[/latex] hours [latex]27[/latex] minutes
7. [latex]12[/latex]
8. [latex]8.618\%[/latex]
9. [latex]$6766.76[/latex]
10. [latex]9[/latex] hours [latex]13[/latex] minutes
11. [latex]239,179[/latex] years
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13. [latex]P\prime (t)=43{e}^{0.01604t}.[/latex] The population is always increasing.
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15. The population reaches [latex]10[/latex] billion people in 2027.
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17. [latex]P\prime (t)=2.259{e}^{0.06407t}.[/latex] The population is always increasing.
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Calculus of the Hyperbolic Functions

1. [latex]{e}^{x}\text{ and }{e}^{\text{−}x}[/latex]
2. Answers may vary
3. Answers may vary
4. Answers may vary
5. [latex]3\text{sinh}(3x+1)[/latex]
6. [latex]\text{−}\text{tanh}(x)\text{sech}(x)[/latex]
7. [latex]4\text{cosh}(x)\text{sinh}(x)[/latex]
8. [latex]\frac{x{\text{sech}}^{2}(\sqrt{{x}^{2}+1})}{\sqrt{{x}^{2}+1}}[/latex]
9. [latex]6{\text{sinh}}^{5}(x)\text{cosh}(x)[/latex]
10. [latex]\frac{1}{2}\text{sinh}(2x+1)+C[/latex]
11. [latex]\frac{1}{2}{\text{sinh}}^{2}({x}^{2})+C[/latex]
12. [latex]\frac{1}{3}{\text{cosh}}^{3}(x)+C[/latex]
13. [latex]\text{ln}(1+\text{cosh}(x))+C[/latex]]
14. [latex]\text{cosh}(x)+\text{sinh}(x)+C[/latex]
15. [latex]\frac{4}{1-16{x}^{2}}[/latex]
16. [latex]\frac{\text{sinh}(x)}{\sqrt{{\text{cosh}}^{2}(x)+1}}[/latex]
17. [latex]\text{−} \csc (x)[/latex]
18. [latex]-\frac{1}{({x}^{2}-1){\text{tanh}}^{-1}(x)}[/latex]
19. [latex]\frac{1}{a}{\text{tanh}}^{-1}(\frac{x}{a})+C[/latex]
20. [latex]\sqrt{{x}^{2}+1}+C[/latex]
21. [latex]{\text{cosh}}^{-1}({e}^{x})+C[/latex]
22. [latex]-0.521095[/latex]
23. [latex]10[/latex]
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