Problem Set: Integrals, Exponential Functions, and Logarithms

For the following exercises (1-3), find the derivative $\frac{dy}{dx}.$

1. $y=\text{ln}(2x)$

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$\frac{1}{x}$

2. $y=\text{ln}(2x+1)$

3. $y=\dfrac{1}{\text{ln}x}$

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$-\frac{1}{x{(\text{ln}x)}^{2}}$

For the following exercises (4-5), find the indefinite integral.

4. $\displaystyle\int \frac{dt}{3t}$

5. $\displaystyle\int \frac{dx}{1+x}$

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$\text{ln}(x+1)+C$

For the following exercises (6-15), find the derivative $dy\text{/}dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

6. [T] $y=\dfrac{\text{ln}(x)}{x}$

7. [T] $y=x\text{ln}(x)$

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$\text{ln}(x)+1$

8. [T] $y={\text{log}}_{10}x$

9. [T] $y=\text{ln}( \sin x)$

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$\cot (x)$

10. [T] $y=\text{ln}(\text{ln}x)$

11. [T] $y=7\text{ln}(4x)$

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$\frac{7}{x}$

12. [T] $y=\text{ln}({(4x)}^{7})$

13. [T] $y=\text{ln}( \tan x)$

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$\csc (x) \sec x$

14. [T] $y=\text{ln}( \tan (3x))$

15. [T] $y=\text{ln}({ \cos }^{2}x)$

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$-2 \tan x$

For the following exercises (16-25), find the definite or indefinite integral.

16. ${\displaystyle\int }_{0}^{1}\dfrac{dx}{3+x}$

17. ${\displaystyle\int }_{0}^{1}\dfrac{dt}{3+2t}$

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$\frac{1}{2}\text{ln}(\frac{5}{3})$

18. ${\displaystyle\int }_{0}^{2}\dfrac{xdx}{{x}^{2}+1}$

19. ${\displaystyle\int }_{0}^{2}\dfrac{{x}^{3}dx}{{x}^{2}+1}$

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$2-\frac{1}{2}\text{ln}(5)$

20. ${\displaystyle\int }_{2}^{e}\dfrac{dx}{x\text{ln}x}$

21. ${\displaystyle\int }_{2}^{e}\dfrac{dx}{{(x\text{ln}(x))}^{2}}$

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$\frac{1}{\text{ln}(2)}-1$

22. $\displaystyle\int \frac{ \cos xdx}{ \sin x}$

23. ${\displaystyle\int }_{0}^{\pi \text{/}4} \tan xdx$

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\frac{1}{2} ln (2)

$\frac{1}{2}\text{ln}(2)$

24. $\displaystyle\int \cot (3x)dx$

25. $\displaystyle\int \frac{{(\text{ln}x)}^{2}dx}{x}$

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$\frac{1}{3}{(\text{ln}x)}^{3}$

For the following exercises (26-35), compute $dy\text{/}dx$ by differentiating $\text{ln}y.$

26. $y=\sqrt{{x}^{2}+1}$

27. $y=\sqrt{{x}^{2}+1}\sqrt{{x}^{2}-1}$

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$\frac{2{x}^{3}}{\sqrt{{x}^{2}+1}\sqrt{{x}^{2}-1}}$

28. $y={e}^{ \sin x}$

29. $y={x}^{-1\text{/}x}$

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${x}^{-2-(1\text{/}x)}(\text{ln}x-1)$

30. $y={e}^{(ex)}$

31. $y={x}^{e}$

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$e{x}^{e-1}$

32. $y={x}^{(ex)}$

33. $y=\sqrt{x}\sqrt[3]{x}\sqrt[6]{x}$

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34. $y={x}^{-1\text{/}\text{ln}x}$

35. $y={e}^{\text{−}\text{ln}x}$

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$-\frac{1}{{x}^{2}}$

For the following exercises (36-40), evaluate by any method.

36. ${\displaystyle\int }_{5}^{10}\frac{dt}{t}-{\displaystyle\int }_{5x}^{10x}\frac{dt}{t}$

37. ${\displaystyle\int }_{1}^{{e}^{\pi }}\frac{dx}{x}+{\displaystyle\int }_{-2}^{-1}\frac{dx}{x}$

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$\pi -\text{ln}(2)$

38. $\frac{d}{dx}{\displaystyle\int }_{x}^{1}\frac{dt}{t}$

39. $\frac{d}{dx}{\displaystyle\int }_{x}^{{x}^{2}}\frac{dt}{t}$

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$\frac{1}{x}$

40. $\frac{d}{dx}\text{ln}( \sec x+ \tan x)$

For the following exercises (41-, use the function $\text{ln}x.$ If you are unable to find intersection points analytically, use a calculator.

41. Find the area of the region enclosed by $x=1$ and $y=5$ above $y=\text{ln}x.$

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${e}^{5}-6{\text{units}}^{2}$

42. [T] Find the arc length of $\text{ln}x$ from $x=1$ to $x=2.$

43. Find the area between $\text{ln}x$ and the $x$ -axis from $x=1\text{ to }x=2.$

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$\text{ln}(4)-1{\text{units}}^{2}$

44. Find the volume of the shape created when rotating this curve from $x=1\text{ to }x=2$ around the $x$ -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 $x=1$ to $x=2$ around the $x$ -axis.

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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 $x=2\text{ and }y=2$ above $y=\frac{1}{x}.$

47. [T] Find the arc length of $y=\frac{1}{x}$ from $x=1\text{ to }x=4.$

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48. Find the area under $y=\frac{1}{x}$ and above the $x$ -axis from $x=1\text{ to }x=4.$

For the following exercises (49-53), verify the derivatives and antiderivatives.

49. $\frac{d}{dx}\text{ln}(x+\sqrt{{x}^{2}+1})=\dfrac{1}{\sqrt{1+{x}^{2}}}$

50. $\frac{d}{dx}\text{ln}\left(\dfrac{x-a}{x+a}\right)=\dfrac{2a}{({x}^{2}-{a}^{2})}$

51. $\frac{d}{dx}\text{ln}\left(\dfrac{1+\sqrt{1-{x}^{2}}}{x}\right)=-\dfrac{1}{x\sqrt{1-{x}^{2}}}$

52. $\frac{d}{dx}\text{ln}(x+\sqrt{{x}^{2}-{a}^{2}})=\dfrac{1}{\sqrt{{x}^{2}-{a}^{2}}}$

53. $\displaystyle\int \frac{dx}{x\text{ln}(x)\text{ln}(\text{ln}x)}=\text{ln}(\text{ln}(\text{ln}x))+C$

Applications of Integration: Supplemental Content

Problem Set: Integrals, Exponential Functions, and Logarithms

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