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Question

Integrate 0π2logsinxdx.


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Solution

Integration of the given Expression:

0afxdx=0afa-xdx.

02afxdx=0afxdx+0af2a-xdx

Let I=0π2logsinxdx...i

By property 1

I=0π2logsinπ2-xdxsinπ2-x=cosxI=0π2logcosxdx...iii+ii2I=0π2logcosx+logsinxdxlogab=loga+logb2I=0π2logcosxsinxdxsin2x=2sinxcosx2I=0π2logsin2x2dx

Let 2x=t

Differentiate both sides

2dx=dt Replacing the value of dx

2I=120πlogsint2dt

By property 2

0πlogsint2dt=0π2logsint2dt+0π2logsinπ-t2dtsinπ-t=sint2I=120π22logsint2dt2I=0π2logsint-0π2log2logab=loga-logb2I=I-π2log2I=-π2log2

Hence 0π2logsinx is -π2log2.


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