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Evaluate the following : ∫-π2π2log(2+sinx2-sinx)⋅dx - Mathematics and Statistics

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Sum

Evaluate the following : `int_((-pi)/2)^(pi/2) log((2 + sinx)/(2 - sinx))*dx`

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Solution

Let I = `int_((-pi)/2)^(pi/2) log((2 - sinx)/(2 + sinx))*dx`

Let f(x) = `log((2 - sinx)/(2 + sinx))`

∴ f(– x)= `log[(2 - sin(-x))/(2 + sin(-x))]` 

= `log((2 + sinx)/(2 - sinx))`

= `-log((2 - sinx)/(2 + sinx))`

= – f(x)

∴ f is an odd function.

∴ `int_((-pi)/2)^(pi/2) f(x)*dx` = 0

∴ `int_((-pi)/2)^(pi/2)log((2 - sinx)/(2 + sinx))*dx` = 0.

Concept: Fundamental Theorem of Integral Calculus
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