Question

Evaluate the integral: \[ \int_0^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x} \, dx \]

Hint:

When dealing with symmetric integrals, use substitution and symmetry properties to simplify the evaluation.

Answer 1

Step 1: Symmetry of the integral.
Observe the symmetry in the integrand. By substituting \( x = \frac{\pi}{2} - t \), we can transform this integral into a simpler form.

Step 2: Simplifying the integral.
Use the symmetry property of definite integrals and trigonometric identities to simplify the integral: \[ \int_0^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x} \, dx = \int_0^{\pi/2} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} \, dx. \]

Step 3: Combining both integrals.
By adding the two integrals, we get: \[ I = \int_0^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x} \, dx + \int_0^{\pi/2} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} \, dx = \frac{\pi}{2}. \] Thus, each integral equals half of \( \frac{\pi}{2} \), which is \( \frac{\pi}{4} \).

Step 4: Conclusion.
Therefore, the value of the integral is \( \frac{\pi}{4} \).