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Question Number 163134 by mnjuly1970 last updated on 04/Jan/22
      prove  or disprove        ∫_(2π) ^( 4π) (( sin(x))/x) dx >0         because   ∫_(2π) ^( 3π) (( sin(x ))/x) dx > ∫_(3π) ^( 4π) ((∣sin(x)∣)/x) dx
$$ \\ $$$$\:\:\:\:{prove}\:\:{or}\:{disprove} \\ $$$$ \\ $$$$\:\:\:\:\int_{\mathrm{2}\pi} ^{\:\mathrm{4}\pi} \frac{\:{sin}\left({x}\right)}{{x}}\:{dx}\:>\mathrm{0} \\ $$$$\:\:\:\:\:\:\:{because} \\ $$$$\:\int_{\mathrm{2}\pi} ^{\:\mathrm{3}\pi} \frac{\:{sin}\left({x}\:\right)}{{x}}\:{dx}\:>\:\int_{\mathrm{3}\pi} ^{\:\mathrm{4}\pi} \frac{\mid{sin}\left({x}\right)\mid}{{x}}\:{dx} \\ $$$$ \\ $$
Answered by mindispower last updated on 04/Jan/22
=∫_0 ^(2π) ((sin(x))/(x+2π))dx=∫_0 ^π ((sin(x))/(x+2π))dx+∫_0 ^π ((−sin(x))/(x+3π))dx  =∫_0 ^π ((πsin(x))/((x+2π)(x+3π)))dx>0  ∫_(3π) ^(4π) ((∣sin(x)∣)/x)dx=∫_0 ^π ((∣sin(x)∣)/(4π−x))dx  ∫_(2π) ^(3π) ((sin(x))/x)dx=∫_0 ^π ((sin(x))/(x+2π))dx  ∫_0 ^π ((sin(x))/(x+2π))dx>∫_0 ^π ((sin(x))/(4π−x))dx...(E) True  x+2π<4π−x true x<π
$$=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{{sin}\left({x}\right)}{{x}+\mathrm{2}\pi}{dx}=\int_{\mathrm{0}} ^{\pi} \frac{{sin}\left({x}\right)}{{x}+\mathrm{2}\pi}{dx}+\int_{\mathrm{0}} ^{\pi} \frac{−{sin}\left({x}\right)}{{x}+\mathrm{3}\pi}{dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi} \frac{\pi{sin}\left({x}\right)}{\left({x}+\mathrm{2}\pi\right)\left({x}+\mathrm{3}\pi\right)}{dx}>\mathrm{0} \\ $$$$\int_{\mathrm{3}\pi} ^{\mathrm{4}\pi} \frac{\mid{sin}\left({x}\right)\mid}{{x}}{dx}=\int_{\mathrm{0}} ^{\pi} \frac{\mid{sin}\left({x}\right)\mid}{\mathrm{4}\pi−{x}}{dx} \\ $$$$\int_{\mathrm{2}\pi} ^{\mathrm{3}\pi} \frac{{sin}\left({x}\right)}{{x}}{dx}=\int_{\mathrm{0}} ^{\pi} \frac{{sin}\left({x}\right)}{{x}+\mathrm{2}\pi}{dx} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{sin}\left({x}\right)}{{x}+\mathrm{2}\pi}{dx}>\int_{\mathrm{0}} ^{\pi} \frac{{sin}\left({x}\right)}{\mathrm{4}\pi−{x}}{dx}…\left({E}\right)\:{True} \\ $$$${x}+\mathrm{2}\pi<\mathrm{4}\pi−{x}\:{true}\:{x}<\pi \\ $$$$ \\ $$
Commented by mnjuly1970 last updated on 04/Jan/22
   grateful sir power  perfect
$$\:\:\:{grateful}\:{sir}\:{power}\:\:{perfect} \\ $$
Commented by mindispower last updated on 04/Jan/22
pleasur sir have nice day
$${pleasur}\:{sir}\:{have}\:{nice}\:{day} \\ $$

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