Evaluate the following integral: ∫ (tan^7x)/(tan^7x + cot^7 x)dx, x ∈ [0, π/2]

Question

Evaluate the following integral:

\(\int\limits_{0}^{\pi/2}\cfrac{tan^7\text x}{tan^7\text x+cot^7\text x}d\text x \)

Answer 1

Let us assume

I = \(\int\limits_{0}^{\pi/2}\cfrac{tan^7\text x}{tan^7\text x+cot^7\text x}d\text x \)...equation 1

By property, we know that

We know

Thus substituting the values in equation 1

Adding equation 1 and equation 2