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```        How can i integrate logsin x from o to pi?

```
7 years ago

Rinkoo Gupta
80 Points
```										Integral 0 to pi (log sinx) dx=2 integral lim 0 to pi/2 ( log sinx) dx using property of definite integralNow let I= integral lim 0 to pi/2 log sinx dx   eq no (1)=integral lim 0 to pi/2 log sin(pi/2-x) dx=integral lim 0 to pi/e log cosx dx           eq no(2)Adding eq (1) and (2) we get2I=integral lom 0 to pi/2 (log sinx + log cosx) dx=> 2I=integral lim 0 to pi/2 log sinx.cosx dx=> 2I=integral lim 0 to pi/2 log (sinxcosx) dx=> 2I=integral lim 0 to pi/2 log ( 2sinxcosx/2) dx=> 2I=integral 0 to pi/2 log sin2x dx - integral 0 to pi/2 log2 dxlet 2x=t in first integral => 2dx=dt and new lim are 0 to pi=> 2I=1/2integral 0 to pi log sint dt -pi/2log2=>2I=1/2 .2 integral 0 to pi/2 log sinx dx -pi/2 log 2 using properties of definite integral=> 2I=integral 0 to pi/2 log sinx dx -pi/2 log2=>2I=I -pi/2 log 2=>I=-pi/2 log 2Hence The value of integral 0 to pi (log sinx) dx=2(-pi/2 log 2)=-pi log2 Ans.Thanks & RegardsRinkoo GuptaaskIItians Faculty
```
3 years ago
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