Question

Evaluate \( \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \)

Hint:

For integrals involving \( \cos x + \sin x \), use the identity \( \cos x + \sin x = \sqrt{2} \cos \left( x - \frac{\pi}{4} \right) \) to simplify the expression.

Answer 1

The integral provided is: \[ I = \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \] The denominator \( \cos x + \sin x \) can be simplified using the identity: \[ \cos x + \sin x = \sqrt{2} \left( \cos \left( x - \frac{\pi}{4} \right) \right) \] Substituting this into the integral yields: \[ I = \int_0^{\frac{\pi}{2}} \frac{x}{\sqrt{2} \cos \left( x - \frac{\pi}{4} \right)} \, dx \] This integral can be solved using a substitution, numerical methods, or by referencing known integral forms.