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int(0)^(pi//2)(dx)/((1+cot^(3)x))=(pi)/(...

`int_(0)^(pi//2)(dx)/((1+cot^(3)x))=(pi)/(4)`

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`I=int_(0)^(pi//2)(dx)/((1+(cos^(3)x)/(sin^(3)x)))=int_(0)^(pi//2)(sin^(3)x)/((sin^(3)x+cos^(3)x))dx`
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int_(0)^(pi//2)(dx)/((1+cotx))=(pi)/(4)

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int_(0)^(pi//4) (dx)/(1+x^(2))

int_(0)^((pi)/(2))(dx)/(1+sqrt(tan x))=int_(0)^((pi)/(2))(dx)/(1+sqrt(cot x))=(pi)/(4)