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Question Number 187458 by mnjuly1970 last updated on 17/Feb/23

$$$$$$$$$$\mathrm{\Omega }={\int }_0^{\frac{\pi }{2}}\frac{2sin\left(x\right)-cos\left(x\right)}{sin\left(x\right)+cos\left(x\right)}dx=?$$$$----$$

Answered by Frix last updated on 17/Feb/23

$$\frac{2\sin x-\cos x}{\sin x+\cos x}-\frac{1}{2}+\frac{1}{2}=$$$$=\frac{3(\sin x-\cos x)}{2(\sin x+\cos x)}+\frac{1}{2}=$$$$=\frac{1}{2}-\frac{3\sqrt{2}\cos (x+\frac{\pi }{4})}{2\sqrt{2}\sin (x+\frac{\pi }{4})}$$$$\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}(\frac{1}{2}-\frac{3}{2}\cot (x+\frac{\pi }{4}))dx=$$$$={[\frac{x}{2}-\frac{3\ln \mid \sin (x+\frac{\pi }{4})\mid }{2}]}_0^{\frac{\pi }{2}}=\frac{\pi }{4}$$

Commented by mnjuly1970 last updated on 17/Feb/23

$$thanksalot$$

Answered by anurup last updated on 17/Feb/23

$$\mathrm{\Omega }=\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}\frac{2\sin x-\cos x}{\sin x+\cos x}dx-\left(1\right)$$$$=\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}\frac{2\sin (\frac{\pi }{2}-x)-\cos (\frac{\pi }{2}-x)}{\sin (\frac{\pi }{2}-x)+\cos (\frac{\pi }{2}-x)}dx$$$$=\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}\frac{2\cos x-\sin x}{\cos x+\sin x}dx-\left(2\right)$$$$\left(1\right)+\left(2\right)\mathrm{gives}$$$$2\mathrm{\Omega }=\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}dx\Rightarrow \mathrm{\Omega }=\frac{\pi }{4}$$

Commented by mnjuly1970 last updated on 17/Feb/23

$$thankyousomuch$$

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