0-pi-2-2sin-x-cos-x-sin-x-cos-x-dx-

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

Ω = \int_{0}^{\frac{π}{2}} \frac{2 s i n (x) - c o s (x)}{s i n (x) + c o s (x)} d x = ?

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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{π}{4})}{2 \sqrt{2} \sin (x + \frac{π}{4})}

\underset{0}{\int^{\frac{π}{2}}} (\frac{1}{2} - \frac{3}{2} \cot (x + \frac{π}{4})) d x =

= {[\frac{x}{2} - \frac{3 \ln ∣ \sin (x + \frac{π}{4}) ∣}{2}]}_{0}^{\frac{π}{2}} = \frac{π}{4}

Commented by mnjuly1970 last updated on 17/Feb/23

t h a n k s a l o t

Answered by anurup last updated on 17/Feb/23

Ω = \underset{0}{\int^{\frac{π}{2}}} \frac{2 \sin x - \cos x}{\sin x + \cos x} d x - (1)

= \underset{0}{\int^{\frac{π}{2}}} \frac{2 \sin (\frac{π}{2} - x) - \cos (\frac{π}{2} - x)}{\sin (\frac{π}{2} - x) + \cos (\frac{π}{2} - x)} d x

= \underset{0}{\int^{\frac{π}{2}}} \frac{2 \cos x - \sin x}{\cos x + \sin x} d x - (2)

(1) + (2) gives

2 Ω = \underset{0}{\int^{\frac{π}{2}}} d x \Rightarrow Ω = \frac{π}{4}

Commented by mnjuly1970 last updated on 17/Feb/23

t h a n k y o u s o m u c h