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I-1-0-pi-sin-884x-sin-1122x-sin-x-dx-I-2-0-1-x-238-x-1768-1-x-2-1-dx-then-value-of-I-1-I-2-




Question Number 39341 by rahul 19 last updated on 05/Jul/18
I_1 = ∫_0 ^π ((sin 884x sin 1122x)/(sin x)) dx  I_2 = ∫_0 ^1  ((x^(238) (x^(1768) −1))/((x^2 −1))) dx  then value of (I_1 /I_2 ) =?
$$\mathrm{I}_{\mathrm{1}} =\:\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{sin}\:\mathrm{884}{x}\:\mathrm{sin}\:\mathrm{1122}{x}}{\mathrm{sin}\:{x}}\:{dx} \\ $$$$\mathrm{I}_{\mathrm{2}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}^{\mathrm{238}} \left({x}^{\mathrm{1768}} −\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)}\:{dx} \\ $$$${then}\:{value}\:{of}\:\frac{\mathrm{I}_{\mathrm{1}} }{\mathrm{I}_{\mathrm{2}} }\:=? \\ $$
Commented by rahul 19 last updated on 06/Jul/18
???????
$$??????? \\ $$
Commented by rahul 19 last updated on 06/Jul/18
pls help.....
$$\mathrm{pls}\:\mathrm{help}….. \\ $$
Commented by ajfour last updated on 06/Jul/18
see Q.39381 (i′ve answered)
$${see}\:{Q}.\mathrm{39381}\:\left({i}'{ve}\:{answered}\right) \\ $$
Commented by abdo mathsup 649 cc last updated on 07/Jul/18
I_2 =∫_0 ^1   ((x^(238) (  (x^2 )^(884)  −1))/(x^2  −1)) dx  = ∫_0 ^1   ((x^(238) (x^2  −1)Σ_(k=0) ^(883)   x^(2k) )/(x^2  −1))dx  = ∫_0 ^1    Σ_(k=0) ^(883)   x^(2k ++238) dx  =Σ_(k=0) ^(883)      [ (1/(2k +239)) x^(2k +239) ]_0 ^1    ...be continued...  =Σ_(k=0) ^(883)    (1/(2k +239))
$${I}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{238}} \left(\:\:\left({x}^{\mathrm{2}} \right)^{\mathrm{884}} \:−\mathrm{1}\right)}{{x}^{\mathrm{2}} \:−\mathrm{1}}\:{dx} \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{238}} \left({x}^{\mathrm{2}} \:−\mathrm{1}\right)\sum_{{k}=\mathrm{0}} ^{\mathrm{883}} \:\:{x}^{\mathrm{2}{k}} }{{x}^{\mathrm{2}} \:−\mathrm{1}}{dx} \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{883}} \:\:{x}^{\mathrm{2}{k}\:++\mathrm{238}} {dx} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{883}} \:\:\:\:\:\left[\:\frac{\mathrm{1}}{\mathrm{2}{k}\:+\mathrm{239}}\:{x}^{\mathrm{2}{k}\:+\mathrm{239}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:\:\:…{be}\:{continued}… \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{883}} \:\:\:\frac{\mathrm{1}}{\mathrm{2}{k}\:+\mathrm{239}} \\ $$

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