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

Explanation

`I=int_{0}^{frac{pi}{2}} frac{d x}{1+tan ^{3} x}…..(1) `


Let `x rightarrow frac{pi}{2}-x `

`
I=int_{0}^{frac{pi}{2}} frac{d x}{1+tan ^{3}(frac{pi}{2}-x)} `

`
=int_{0}^{frac{pi}{2}} frac{d x}{1+cot ^{3} x} `

`
=int_{0}^{frac{pi}{2}} frac{d x}{1+frac{1}{tan ^{3} x}} `

`
=int_{0}^{frac{pi}{2}} frac{tan ^{3} x d x}{tan ^{3} x+1}`



Merge (1) and (2) we obtain

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