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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 bymnjuly1970 last updated on 04/Jan/22

grateful sir power perfect

$$\:\:\:{grateful}\:{sir}\:{power}\:\:{perfect} \\ $$

Commented bymindispower last updated on 04/Jan/22

pleasur sir have nice day

$${pleasur}\:{sir}\:{have}\:{nice}\:{day} \\ $$

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