Evaluate the following integral: `int_0^(pi//2)1/(1+tan^3x)dx`

Explanation

Given:`int_0^(pi//2)1/(1+tan^3x)dx`

`I=int_0^(pi//2)1/(1+tan^3x)dx......(i)`

Now replace x by `pi/2-x`

`I=int_0^(pi//2)1/(1+tan^3(pi/2-x))dx`

`I=int_0^(pi//2)1/(1+cot^3x)dx`

`I=int_0^(pi//2)1/(1+1/(tan^3x))dx`

`I=int_0^(pi//2)(tan^3x)/(tan^3x+1)dx......(ii)`

Now add (i) and (ii) we get

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