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Getting Natural With Pi (Posted on 2007-11-27) Difficulty: 3 of 5
Determine the value of the constant y, whenever:
       y
     (ex - 1)-0.5 dx = pi/6 
      ln(4/3)

where ln x denotes the natural logarithm of x.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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re(3): There are two solutions | Comment 7 of 8 |
(In reply to re(2): There are two solutions by Kurious)

Numerical program for the integral from ln(4/3) to ln(8-4abs(sqrt(3))):

DEFDBL A-Z
pi = ATN(1) * 4
h = -.0000001
y0 = LOG(4 / 3)
yb = y0
yLast = LOG(8 - 4 * ABS(SQR(3)))
CLS
   x = yb
   t = 0
   DO
     t = t + h / SQR(EXP(x) - 1)
     tprev = t
     x = x + h
   LOOP UNTIL x < yLast
   PRINT x - h, tprev, tprev - pi / 6
   PRINT x, t, t - pi / 6

produces

6.933646990017295D-02      -.5235990275116614          -1.04719780310996
6.933636990017178D-02      -.5235990275116614          -1.04719780310996

showing the integral to be -pi/6 rather than pi/6.

To be positive the integral would have to be from ln(8-4abs(sqrt(3))) to ln(4/3) rather than the other way around. Any definite integral from a to b is the negative of that integral from b to a.

 


  Posted by Charlie on 2007-11-28 09:23:19
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