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

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 which states that $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$. Here, $a = \frac{\pi}{2}$. Applying this property, $\sin(\frac{\pi}{2}-x) = \cos(x)$ and $\cos(\frac{\pi}{2}-x) = \sin(x)$.

Step 3: Simplify the integral

After applying the property, the integral transforms into this new form. We now have two expressions for I.

Step 4: Add the two integral expressions

Adding the original integral (from Step 1) and the transformed integral (from Step 3) gives $2I$. The denominators are the same, so we can combine the numerators.

Step 5: Simplify and evaluate the integral

The numerator and denominator cancel out, leaving an integral of $1$ with respect to $x$. Integrating $1$ gives $x$. Evaluating $x$ from $0$ to $\frac{\pi}{2}$ yields $\frac{\pi}{2}$.

Step 6: Solve for I

Since $2I = \frac{\pi}{2}$, we divide by $2$ to find the value of I.

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