Menu Close

Question-9057




Question Number 9057 by sandipkd@ last updated on 16/Nov/16
Answered by aydnmustafa1976 last updated on 16/Nov/16
nsin(1/n)=lim((sin(1/n))/(1/n))=lim((sint)/t)=1 therefore  4∫_0 ^1 (1/(x^2 +1))dx=4.arctgx∣_0 ^1 =4((Π/4)−0)=Π
$${nsin}\frac{\mathrm{1}}{{n}}={lim}\frac{{sin}\frac{\mathrm{1}}{{n}}}{\frac{\mathrm{1}}{{n}}}={lim}\frac{{sint}}{{t}}=\mathrm{1}\:{therefore}\:\:\mathrm{4}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=\mathrm{4}.{arctgx}\mid_{\mathrm{0}} ^{\mathrm{1}} =\mathrm{4}\left(\frac{\Pi}{\mathrm{4}}−\mathrm{0}\right)=\Pi \\ $$
Commented by sandipkd@ last updated on 17/Nov/16
thanks..i was confused in nsin 1/n
$${thanks}..{i}\:{was}\:{confused}\:{in}\:{n}\mathrm{sin}\:\mathrm{1}/{n} \\ $$
Answered by aydnmustafa1976 last updated on 17/Nov/16
lim _(n→∞) ∫_0 ^(nsin(1/n)) ..... =∫_0 ^(lim(1/(1/n)).sin(1/n)) ....=∫_0 ^(lim((sin(1/n))/(1/n))) ...=....
$${lim}\underset{{n}\rightarrow\infty} {\:}\int_{\mathrm{0}} ^{{nsin}\frac{\mathrm{1}}{{n}}} …..\:=\int_{\mathrm{0}} ^{{lim}\frac{\mathrm{1}}{\frac{\mathrm{1}}{{n}}}.{sin}\frac{\mathrm{1}}{{n}}} ….=\int_{\mathrm{0}} ^{{lim}\frac{{sin}\frac{\mathrm{1}}{{n}}}{\frac{\mathrm{1}}{{n}}}} …=…. \\ $$
Answered by aydnmustafa1976 last updated on 17/Nov/16
note: lim_(n→∞) ((sin(1/n))/(1/n))=li_((1/n)→0) m((sin(1/n))/(1/n))=lim_(t→0) ((sint)/t)=0
$${note}:\:{li}\underset{{n}\rightarrow\infty} {{m}}\frac{{sin}\frac{\mathrm{1}}{{n}}}{\frac{\mathrm{1}}{{n}}}={l}\underset{\frac{\mathrm{1}}{{n}}\rightarrow\mathrm{0}} {{i}m}\frac{{sin}\frac{\mathrm{1}}{{n}}}{\frac{\mathrm{1}}{{n}}}={li}\underset{{t}\rightarrow\mathrm{0}} {{m}}\frac{{sint}}{{t}}=\mathrm{0} \\ $$
Answered by aydnmustafa1976 last updated on 17/Nov/16
sorry lim((sint)/t)=1 when t→0
$${sorry}\:{lim}\frac{{sint}}{{t}}=\mathrm{1}\:{when}\:{t}\rightarrow\mathrm{0} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *