Problem Set: Integrals, Exponential Functions, and Logarithms

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.

This figure is a surface. It has been generated by revolving the curve ln x about the x-axis. The surface is inside of a cube showing it is 3-dimensinal.

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]