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

We are given the integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \] Let’s first simplify the denominator \( \cos x + \sin x \). Using the identity: \[ \cos x + \sin x = \sqrt{2} \left( \cos \left( x - \frac{\pi}{4} \right) \right) \] Substitute this into the integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{x}{\sqrt{2} \cos \left( x - \frac{\pi}{4} \right)} \, dx \] This integral requires a substitution, or one can use numerical methods or known results for integrals of this form.