prove that : ∫_(−(π/2)) ^(−(π/4)) 2cos(x)+sin(x)dx≤∫_(−(π/2)) ^(−(π/4)) cos(x)−sin(x)dx


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Question Number 109082 by 1777 last updated on 21/Aug/20

$p r o v e t h a t :$ $\int_{- \frac{π}{2}}^{- \frac{π}{4}} 2 c o s (x) + s i n (x) d x ⩽ \int_{- \frac{π}{2}}^{- \frac{π}{4}} c o s (x) - s i n (x) d x$
	Answered by malwan last updated on 21/Aug/20

	$- \frac{π}{2} < x < - \frac{π}{4}$ $\Rightarrow 0 < c o s x < \frac{\sqrt{2}}{2}$ $\Rightarrow 0 < 2 c o s x < \sqrt{2}$ $- 1 < s i n x < - \frac{\sqrt{2}}{2}$ $\Rightarrow 2 c o s x + s i n x < \sqrt{2} - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ $c o s x - s i n x < \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}$ $\frac{\sqrt{2}}{2} < \sqrt{2}$ $\Rightarrow 2 c o s x - s i n x < c o s x - s i n x$ $∴_{- \frac{π}{2}} \int^{- \frac{π}{4}} (2 c o s x + s i n x) d x <$ $_{- \frac{π}{2}} \int^{- \frac{π}{4}} (c o s x - s i n x) d x$
	Commented by 1777 last updated on 21/Aug/20
	nice
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