For the following exercises (1-3), find the derivative [latex]\frac{dy}{dx}.[/latex]
1. [latex]y=\text{ln}(2x)[/latex]
2. [latex]y=\text{ln}(2x+1)[/latex]
3. [latex]y=\dfrac{1}{\text{ln}x}[/latex]
For the following exercises (4-5), find the indefinite integral.
4. [latex]\displaystyle\int \frac{dt}{3t}[/latex]
5. [latex]\displaystyle\int \frac{dx}{1+x}[/latex]
For the following exercises (6-15), find the derivative [latex]dy\text{/}dx.[/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)
6. [T] [latex]y=\dfrac{\text{ln}(x)}{x}[/latex]
7. [T] [latex]y=x\text{ln}(x)[/latex]
8. [T] [latex]y={\text{log}}_{10}x[/latex]
9. [T] [latex]y=\text{ln}( \sin x)[/latex]
10. [T] [latex]y=\text{ln}(\text{ln}x)[/latex]
11. [T] [latex]y=7\text{ln}(4x)[/latex]
12. [T] [latex]y=\text{ln}({(4x)}^{7})[/latex]
13. [T] [latex]y=\text{ln}( \tan x)[/latex]
14. [T] [latex]y=\text{ln}( \tan (3x))[/latex]
15. [T] [latex]y=\text{ln}({ \cos }^{2}x)[/latex]
For the following exercises (16-25), find the definite or indefinite integral.
16. [latex]{\displaystyle\int }_{0}^{1}\dfrac{dx}{3+x}[/latex]
17. [latex]{\displaystyle\int }_{0}^{1}\dfrac{dt}{3+2t}[/latex]
18. [latex]{\displaystyle\int }_{0}^{2}\dfrac{xdx}{{x}^{2}+1}[/latex]
19. [latex]{\displaystyle\int }_{0}^{2}\dfrac{{x}^{3}dx}{{x}^{2}+1}[/latex]
20. [latex]{\displaystyle\int }_{2}^{e}\dfrac{dx}{x\text{ln}x}[/latex]
21. [latex]{\displaystyle\int }_{2}^{e}\dfrac{dx}{{(x\text{ln}(x))}^{2}}[/latex]
22. [latex]\displaystyle\int \frac{ \cos xdx}{ \sin x}[/latex]
23. [latex]{\displaystyle\int }_{0}^{\pi \text{/}4} \tan xdx[/latex]
24. [latex]\displaystyle\int \cot (3x)dx[/latex]
25. [latex]\displaystyle\int \frac{{(\text{ln}x)}^{2}dx}{x}[/latex]
For the following exercises (26-35), compute [latex]dy\text{/}dx[/latex] by differentiating [latex]\text{ln}y.[/latex]
26. [latex]y=\sqrt{{x}^{2}+1}[/latex]
27. [latex]y=\sqrt{{x}^{2}+1}\sqrt{{x}^{2}-1}[/latex]
28. [latex]y={e}^{ \sin x}[/latex]
29. [latex]y={x}^{-1\text{/}x}[/latex]
30. [latex]y={e}^{(ex)}[/latex]
31. [latex]y={x}^{e}[/latex]
32. [latex]y={x}^{(ex)}[/latex]
33. [latex]y=\sqrt{x}\sqrt[3]{x}\sqrt[6]{x}[/latex]
34. [latex]y={x}^{-1\text{/}\text{ln}x}[/latex]
35. [latex]y={e}^{\text{−}\text{ln}x}[/latex]
For the following exercises (36-40), evaluate by any method.
36. [latex]{\displaystyle\int }_{5}^{10}\frac{dt}{t}-{\displaystyle\int }_{5x}^{10x}\frac{dt}{t}[/latex]
37. [latex]{\displaystyle\int }_{1}^{{e}^{\pi }}\frac{dx}{x}+{\displaystyle\int }_{-2}^{-1}\frac{dx}{x}[/latex]
38. [latex]\frac{d}{dx}{\displaystyle\int }_{x}^{1}\frac{dt}{t}[/latex]
39. [latex]\frac{d}{dx}{\displaystyle\int }_{x}^{{x}^{2}}\frac{dt}{t}[/latex]
40. [latex]\frac{d}{dx}\text{ln}( \sec x+ \tan x)[/latex]
For the following exercises (41-, use the function [latex]\text{ln}x.[/latex] If you are unable to find intersection points analytically, use a calculator.
41. Find the area of the region enclosed by [latex]x=1[/latex] and [latex]y=5[/latex] above [latex]y=\text{ln}x.[/latex]
42. [T] Find the arc length of [latex]\text{ln}x[/latex] from [latex]x=1[/latex] to [latex]x=2.[/latex]
43. Find the area between [latex]\text{ln}x[/latex] and the [latex]x[/latex]-axis from [latex]x=1\text{ to }x=2.[/latex]
44. Find the volume of the shape created when rotating this curve from [latex]x=1\text{ to }x=2[/latex] around the [latex]x[/latex]-axis, as pictured here.
45. [T] Find the surface area of the shape created when rotating the curve in the previous exercise from [latex]x=1[/latex] to [latex]x=2[/latex] around the [latex]x[/latex]-axis.
If you are unable to find intersection points analytically in the following exercises (46-48), use a calculator.
46. Find the area of the hyperbolic quarter-circle enclosed by [latex]x=2\text{ and }y=2[/latex] above [latex]y=\frac{1}{x}.[/latex]
47. [T] Find the arc length of [latex]y=\frac{1}{x}[/latex] from [latex]x=1\text{ to }x=4.[/latex]
48. Find the area under [latex]y=\frac{1}{x}[/latex] and above the [latex]x[/latex]-axis from [latex]x=1\text{ to }x=4.[/latex]
For the following exercises (49-53), verify the derivatives and antiderivatives.
49. [latex]\frac{d}{dx}\text{ln}(x+\sqrt{{x}^{2}+1})=\dfrac{1}{\sqrt{1+{x}^{2}}}[/latex]
50. [latex]\frac{d}{dx}\text{ln}\left(\dfrac{x-a}{x+a}\right)=\dfrac{2a}{({x}^{2}-{a}^{2})}[/latex]
51. [latex]\frac{d}{dx}\text{ln}\left(\dfrac{1+\sqrt{1-{x}^{2}}}{x}\right)=-\dfrac{1}{x\sqrt{1-{x}^{2}}}[/latex]
52. [latex]\frac{d}{dx}\text{ln}(x+\sqrt{{x}^{2}-{a}^{2}})=\dfrac{1}{\sqrt{{x}^{2}-{a}^{2}}}[/latex]
53. [latex]\displaystyle\int \frac{dx}{x\text{ln}(x)\text{ln}(\text{ln}x)}=\text{ln}(\text{ln}(\text{ln}x))+C[/latex]
