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 Getting Natural With Pi (Posted on 2007-11-27)
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
 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          -1.047197803109966.933636990017178D-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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