The value of
\[
\int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx
\]
is equal to:
Show Hint
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{2} \)
- \( 2\pi \)
- \( \pi \)
- \( \frac{\pi}{3} \)
The Correct Option is A
Solution and Explanation
We are asked to solve the integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] Step 1: Notice that the integrand has symmetry. Let's perform a substitution \( x \to \frac{\pi}{2} - x \).
Under this substitution:
\[ \sin \left( \frac{\pi}{2} - x \right) = \cos x, \quad \cos \left( \frac{\pi}{2} - x \right) = \sin x. \] Step 2: Substituting in the integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} \left( \frac{\pi}{2} - x \right)}{\sin^{2024} \left( \frac{\pi}{2} - x \right) + \cos^{2024} \left( \frac{\pi}{2} - x \right)} \, dx \] This transforms the integral to: \[ I = \int_0^{\frac{\pi}{2}} \frac{\sin^{2024} x}{\cos^{2024} x + \sin^{2024} x} \, dx. \] Step 3: Adding the two equations, we get: \[ 2I = \int_0^{\frac{\pi}{2}} \left( \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} + \frac{\sin^{2024} x}{\sin^{2024} x + \cos^{2024} x} \right) \, dx \] Simplifying the integrand: \[ 2I = \int_0^{\frac{\pi}{2}} 1 \, dx \] \[ 2I = \frac{\pi}{2} \] Step 4: Therefore, \( I = \frac{\pi}{4} \).
Thus, the value of the integral is \( \frac{\pi}{4} \).
Therefore, the correct answer is option (A).
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