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Mathematics LibreTexts

2.7E: Exercises for Section 6.7

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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In exercises 1 - 3, find the derivative dydx.

1) y=ln(2x)

Answer
dydx=1x

2) y=ln(2x+1)

3) y=1lnx

Answer
dydx=1x(lnx)2

In exercises 4 - 5, find the indefinite integral.

4) dt3t

5) dx1+x

Answer
dx1+x=ln|x+1|+C

In exercises 6 - 15, find the derivative dydx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

6) [T] y=lnxx

7) [T] y=xlnx

Answer
dydx=ln(x)+1

8) [T] y=log10x

9) [T] y=ln(sinx)

Answer
dydx=cotx

10) [T] y=ln(lnx)

11) [T] y=7ln(4x)

Answer
dydx=7x

12) [T] y=ln((4x)7)

13) [T] y=ln(tanx)

Answer
dydx=cscxsecx

14) [T] y=ln(tan3x)

15) [T] y=ln(cos2x)

Answer
dydx=2tanx

In exercises 16 - 25, find the definite or indefinite integral.

16) 10dx3+x

17) 10dt3+2t

Answer
10dt3+2t=12ln(53)

18) 20xx2+1dx

19) 20x3x2+1dx

Answer
20x3x2+1dx=212ln(5)

20) e2dxxlnx

21) e2dx(xlnx)2

Answer
e2dx(xlnx)2=1ln(2)1

22) cosxsinxdx

23) π/40tanxdx

Answer
π/40tanxdx=12ln(2)

24) cot(3x)dx

25) (lnx)2xdx

Answer
(lnx)2xdx=13(lnx)3

In exercises 26 - 35, compute dydx by differentiating lny.

26) y=x2+1

27) y=x2+1x21

Answer
dydx=2x3x2+1x21

28) y=esinx

29) y=x1/x

Answer
dydx=x2(1/x)(lnx1)

30) y=eex

31) y=xe

Answer
dydx=exe1

32) y=x(ex)

33) y=x3x6x

Answer
dydx=1

34) y=x1/lnx

35) y=elnx

Answer
dydx=1x2

In exercises 36 - 40, evaluate by any method.

36) 105dtt10x5xdtt

37) eπ1dxx+12dxx

Answer
πln(2)

38) ddx[1xdtt]

39) ddx[x2xdtt]

Answer
1x

40) ddx[ln(secx+tanx)]

In exercises 41 - 44, use the function lnx. 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=lnx.

Answer
(e56) units2

42) [T] Find the arc length of lnx from x=1 to x=2.

43) Find the area between lnx and the x-axis from x=1 to x=2.

Answer
ln(4)1) units2

44) Find the volume of the shape created when rotating this curve from x=1 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.

Answer
2.8656 units2

If you are unable to find intersection points analytically in the following exercises, use a calculator.

46) Find the area of the hyperbolic quarter-circle enclosed by x=2 and y=2 above y=1/x.

47) [T] Find the arc length of y=1/x from x=1 to x=4.

Answer
s=3.1502 units

48) Find the area under y=1/x and above the x-axis from x=1 to x=4.

In exercises 49 - 53, verify the derivatives and antiderivatives.

49) ddx[ln(x+x2+1)]=11+x2

50) ddx[ln(xax+a)]=2a(x2a2)

51) ddx[ln(1+1x2x)]=1x1x2

52) ddx[ln(x+x2a2)]=1x2a2

53) dxxln(x)ln(lnx)=ln|ln(lnx)|+C

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


This page titled 2.7E: Exercises for Section 6.7 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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