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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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analytic solution | Comment 2 of 3 |

integral sin(x)*ln(sin(x)) dx
using integration by parts with
u=ln(sin(x)) and dv=sin(x) dx then we have
du=cot(x) dx and v=-cos(x) thus we have
integral sin(x)*ln(sin(x)) dx=
-cos(x)*ln(sin(x))+integral cos(x)*cot(x) dx=
-cos(x)*ln(sin(x))+integral cos^2(x)/sin(x) dx
now cos^2(x)=1-sin^2(x) so we now have
-cos(x)*ln(sin(x))+integral (1-sin^2(x))/sin(x) dx=
-cos(x)*ln(sin(x))+integral csc(x)-sin(x) dx=
-cos(x)*ln(sin(x))-ln(csc(x)+cot(x))+cos(x)
evaluating this on the interval 0 to 1 we get
ln(2)-1 or approximately -0.306853 which agrees with Charlie's numeric solution

Edited on May 19, 2009, 1:06 pm
  Posted by Daniel on 2009-05-19 13:04:15

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