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Getting Natural With Sine (Posted on 2009-05-19) Difficulty: 2 of 5
Evaluate:

pi/2
sin y*ln(sin y) dy
0

where, ln y denotes the natural logarithm of y.

No Solution Yet Submitted by K Sengupta    
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Solution Using Wolfram's Online Integrator Comment 3 of 3 |
(In reply to numerical integration with computer--but what is this number? by Charlie)

Using Wolfram's Online Integrator, you have to use x rather than y and ln is just log, but, converted to use of y and ln, the antiderivative given there is:

-ln(sin(y)) * cos(y) + cos(y) + ln(tan(y/2))

Evaluated at y = pi/2, this comes out to just zero.

But evaluated at y = 0, the first of the three terms would become positive infinity and the last term negative infinity, making the middle term of 1 useless.

One would have to seek the limit of this as y approaches zero.

cos(y) approaches 1 obviously.

ln(tan(y/2)) - ln(sin(y)) = ln(tan(y/2)/sin(y))

Since tan(y)/sin(y) approaches 1, tan(y/2)/sin(y) approaches 1/2.

So the definite integral would be 0 - (1 + ln(1/2)) = -1 - ln(1/2) = ln(2) - 1

ln(2) ~= .6931471805599453

so the integral does evaluate to  -0.3068528194400547


  Posted by Charlie on 2009-05-19 13:09:31
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