Evaluate the integral from 0 to pi/2 of (sin(x))^5 / ((sin(x))^5 + (cos(x))^5) dx.

Step 1: Define the integral

Let the given integral be denoted by I. This is the standard way to begin solving definite integral problems.

Step 2: Apply the property of definite integrals

We use the property of definite integrals that states $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ $. Here, a = \frac{\pi}{2}$. We substitute x with ($\frac{\pi}{2}-x$) in the integrand.

Step 3: Simplify the integrand

Using the trigonometric identities $\sin(\frac{\pi}{2}-x) = \cos x$ and $\cos(\frac{\pi}{2}-x) = \sin x$, we simplify the integrand. This gives us a new expression for I.

Step 4: Add the original and modified integrals

We add the original integral (from Step 1) and the modified integral (from Step 3). Since both integrals have the same limits and the same denominator in their integrands, we can combine them.

Step 5: Simplify and evaluate

The numerator and denominator of the integrand cancel out, leaving us with $\int_{0}^{\frac{\pi}{2}} 1$ \, dx. The integral of 1 with respect to x is x. Evaluating this from 0 to $\frac{\pi}{2}$ gives $\frac{\pi}{2}$. Finally, we solve for I by dividing by 2.

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