Mathematics - Definite Integration Question with Solution | TestHub

Let $I={\int }_0^{\frac{\pi }{2}}\frac{\sin \left(\frac{\pi }{4}+x\right)+\sin \left(\frac{3\pi }{4}+x\right)}{\cos x+\sin x}dx$ $\Rightarrow I={\int }_0^{\pi /2}\frac{\sin \left(\frac{3\pi }{4}-x\right)+\sin \left(\frac{5\pi }{4}-x\right)}{\sin x+\cos x}dx$ $\Rightarrow I={\int }_0^{\pi /2}\frac{\sin \left(\frac{\pi }{4}+x\right)-\sin \left(\frac{3\pi }{4}+x\right)}{\cos x+\sin x}dx$ Now $I+I={\int }_0^{\pi /2}\frac{2\sin \left(\frac{\pi }{4}+x\right)}{\cos x+\sin x}dx$ $\Rightarrow I={\int }_0^{\pi /2}\frac{\frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x}{\cos x-\sin x}dx=\frac{\pi }{2\sqrt{2}}$