Question Number 173372 by Shrinava last updated on 10/Jul/22
Answered by a.lgnaoui last updated on 11/Jul/22
![(1/((1+x^2 )))×(1/((1+xlog^7 x)))=(1/((1+xlog^7 x)))−(x^2 /((x^2 +1)(1+xlog^7 x))) =(1/(1+xlog^7 x))−(x^2 /(x^2 (1+(1/x^2 ))(1+xlog^7 x)))= (1/(1+xlog^7 x))−(1/((1+(1/x^2 ))(1+xlog^7 x)))= (1/(1+xlog^7 x))(1−(1/(1+(1/x^2 ))))=(1/(1+xlog^7 x))(1−(1/(x^2 +1))) tel I=∫(dx/((x^2 +1)(1+xlog^7 x))) I=∫(1/(1+xlog^7 x))−I ⇒2I=∫(1/(1+xlog^7 x))dx I=(1/2)∫(1/(1+xlog^7 x))dx tel u′=1 and v=(1/(1+xlog^7 x)) ⇒u=x v′=((−(log^7 x+x×7×(1/x)log^6 x))/((1+xlog^7 x)^2 )) =((−log^6 x(7+logx))/((1+xlog^7 x)^2 )) ;∫(1/(1+xlog^7 x))=[(x/(1+xlog^7 x))]+∫x(((7log^6 x+log^7 x))/((1+xlg^7 x)^2 )) ∫((7log^6 x+log^7 x)/((1+xlog^7 x)))xdx=∫((log^7 x((7/(logx))+1)x)/(x^2 log^(14) x((1/(xlog^7 x))+1)^2 ))⇒ ∫(1/(1+xlog^7 x))=[(x/(1+xlog^7 x))]+∫((((7/(logx))+1))/(xlog^7 x(((1+xlog^7 x)^2 )/(x^2 log^(14) x))))) =[(x/(1+xlog^7 x))]+∫(((7+logx)(xlog^6 x))/((1+xlog^7 x)^2 )) =[(x/(1+xlog^7 x))]+∫((x(log^7 x+7log^6 x))/((1+xlog^7 x)^2 )) let t=xlog^7 x or t=xlog^7 x dt/dx=log^7 x+log^6 x so []+∫((7dt)/((1+t)^2 ))−6∫(t/((1+t)^2 ))dt 2I=[(x/(1+xlog^7 x))]_(1/e) ^e +∫_(−(1/e)) ^(1/e) ((7dt−6t)/((1+t)^2 ))dt 7dt−6t=−(6t+6dt)+13dt and ∫ =[[−∫(dt/(1+t))+13∫(dt/((1+t)^2 )) I=(1/2)[(x/(1+xlog^7 x))]+[log(1+t)]_((−1)/e) ^(1/e) +13∫_(−(1/e)) ^(1/e) (dt/((1+t)^2 )) I=∫(1/(1+xlog^7 x))=∫(1/(1+t))=(1/2)[(x/(1+xlog^7 x))]_(1/e) ^(1/e) +(1/2)[log(1+t)]_(−(1/e)) ^(1/e) −13[(1/(1+t))]_(−1/e) ^(1/e)](https://www.tinkutara.com/question/Q173451.png)
$$\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}×\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)}=\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)}−\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)}= \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}−\frac{\mathrm{1}}{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)}= \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}\right)=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\right) \\ $$$$\mathrm{tel}\:\:\mathrm{I}=\int\frac{{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)}\:\: \\ $$$$\mathrm{I}=\int\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}−\mathrm{I}\:\:\:\Rightarrow\mathrm{2I}=\int\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\mathrm{dx}\:\:\:\mathrm{I}=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\mathrm{dx} \\ $$$$\mathrm{tel}\:\:\:\:\mathrm{u}'=\mathrm{1}\:\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{v}=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\:\:\:\Rightarrow{u}=\mathrm{x}\:\:\:\:\mathrm{v}'=\frac{−\left(\mathrm{log}^{\mathrm{7}} \mathrm{x}+\mathrm{x}×\mathrm{7}×\frac{\mathrm{1}}{\mathrm{x}}\mathrm{log}^{\mathrm{6}} \mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)^{\mathrm{2}} } \\ $$$$=\frac{−\mathrm{log}^{\mathrm{6}} \mathrm{x}\left(\mathrm{7}+\mathrm{logx}\right)}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)^{\mathrm{2}} }\:\:;\int\frac{\mathrm{1}}{\mathrm{1}+{x}\mathrm{log}^{\mathrm{7}} \mathrm{x}}=\left[\frac{\mathrm{x}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\right]+\int\mathrm{x}\frac{\left(\mathrm{7log}^{\mathrm{6}} \mathrm{x}+\mathrm{log}^{\mathrm{7}} \mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{xlg}^{\mathrm{7}} \mathrm{x}\right)^{\mathrm{2}} } \\ $$$$\int\frac{\mathrm{7log}^{\mathrm{6}} \mathrm{x}+\mathrm{log}^{\mathrm{7}} \mathrm{x}}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)}\mathrm{xdx}=\int\frac{\mathrm{log}^{\mathrm{7}} \mathrm{x}\left(\frac{\mathrm{7}}{\mathrm{logx}}+\mathrm{1}\right)\mathrm{x}}{\mathrm{x}^{\mathrm{2}} \mathrm{log}^{\mathrm{14}} \mathrm{x}\left(\frac{\mathrm{1}}{\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}+\mathrm{1}\right)^{\mathrm{2}} }\Rightarrow \\ $$$$\int\frac{\mathrm{1}}{\mathrm{1}+{x}\mathrm{log}^{\mathrm{7}} \mathrm{x}}=\left[\frac{\mathrm{x}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\right]+\int\frac{\left(\frac{\mathrm{7}}{\mathrm{logx}}+\mathrm{1}\right)}{\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\frac{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)^{\mathrm{2}} }{\left.\mathrm{x}^{\mathrm{2}} \mathrm{log}^{\mathrm{14}} \mathrm{x}\right)}} \\ $$$$=\left[\frac{\mathrm{x}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\right]+\int\frac{\left(\mathrm{7}+\mathrm{logx}\right)\left(\mathrm{xlog}^{\mathrm{6}} \mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)^{\mathrm{2}} } \\ $$$$=\left[\frac{\mathrm{x}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\right]+\int\frac{{x}\left(\mathrm{log}^{\mathrm{7}} \mathrm{x}+\mathrm{7log}^{\mathrm{6}} \mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\right)^{\mathrm{2}} } \\ $$$$\mathrm{let}\:\:\:\:\:\mathrm{t}=\mathrm{xlog}^{\mathrm{7}} \mathrm{x}\:\:\: \\ $$$${or}\:{t}={xlog}^{\mathrm{7}} {x}\:\:\:{dt}/{dx}={log}^{\mathrm{7}} {x}+{log}^{\mathrm{6}} {x} \\ $$$$\mathrm{so}\:\:\:\left[\right]+\int\frac{\mathrm{7}{dt}}{\left(\mathrm{1}+\mathrm{t}\right)^{\mathrm{2}} }−\mathrm{6}\int\frac{{t}}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$$$\mathrm{2I}=\left[\frac{\mathrm{x}}{\mathrm{1}+\mathrm{xlog}^{\mathrm{7}} \mathrm{x}}\right]_{\frac{\mathrm{1}}{{e}}} ^{{e}} +\int_{−\frac{\mathrm{1}}{{e}}} ^{\frac{\mathrm{1}}{{e}}} \frac{\mathrm{7}{dt}−\mathrm{6}{t}}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$$$\mathrm{7}{dt}−\mathrm{6}{t}=−\left(\mathrm{6}{t}+\mathrm{6}{dt}\right)+\mathrm{13}{dt} \\ $$$${and}\:\int\:\:\:\:\:\:=\left[\left[−\int\frac{{dt}}{\mathrm{1}+{t}}+\mathrm{13}\int\frac{{dt}}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }\right.\right. \\ $$$$\mathrm{I}=\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{{x}}{\mathrm{1}+{xlog}^{\mathrm{7}} {x}}\right]+\left[\mathrm{log}\left(\mathrm{1}+\mathrm{t}\right)\right]_{\frac{−\mathrm{1}}{{e}}} ^{\frac{\mathrm{1}}{{e}}} +\mathrm{13}\int_{−\frac{\mathrm{1}}{{e}}} ^{\frac{\mathrm{1}}{{e}}} \frac{{dt}}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} } \\ $$$$\mathrm{I}=\int\frac{\mathrm{1}}{\mathrm{1}+{x}\mathrm{log}^{\mathrm{7}} \mathrm{x}}=\int\frac{\mathrm{1}}{\mathrm{1}+{t}}=\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{{x}}{\mathrm{1}+{x}\mathrm{log}^{\mathrm{7}} \mathrm{x}}\right]_{\frac{\mathrm{1}}{{e}}} ^{\frac{\mathrm{1}}{{e}}} +\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{log}\left(\mathrm{1}+\mathrm{t}\right)\right]_{−\frac{\mathrm{1}}{{e}}} ^{\frac{\mathrm{1}}{{e}}} −\mathrm{13}\left[\frac{\mathrm{1}}{\mathrm{1}+{t}}\right]_{−\mathrm{1}/{e}} ^{\mathrm{1}/{e}} \\ $$
Commented by Shrinava last updated on 12/Jul/22
$$\mathrm{thankyou}\:\mathrm{professor},\:\mathrm{but}\:\mathrm{ansver}? \\ $$
Commented by Tawa11 last updated on 13/Jul/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Commented by a.lgnaoui last updated on 13/Jul/22
![I=(1/2)[(x/(1+xln^7 x))]_(1/e) ^e +(1/2)[log(1+xlog^7 x)]_(1/e) ^e −13[(1/(1+xln^7 x))]_(1/e) ^e =((e−1)/(2(e+1)))+(1/2)[log^7 (((e+1)/(e−1)))]+((26)/(e^2 −1))](https://www.tinkutara.com/question/Q173563.png)
$$\mathrm{I}=\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{{x}}{\mathrm{1}+{x}\mathrm{ln}^{\mathrm{7}} {x}}\right]_{\frac{\mathrm{1}}{{e}}} ^{{e}} +\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{log}\left(\mathrm{1}+{x}\mathrm{l}{og}^{\mathrm{7}} {x}\right)\right]_{\frac{\mathrm{1}}{{e}}} ^{{e}} −\mathrm{13}\left[\frac{\mathrm{1}}{\mathrm{1}+{x}\mathrm{ln}^{\mathrm{7}} {x}}\right]_{\frac{\mathrm{1}}{{e}}} ^{{e}} \\ $$$$=\frac{{e}−\mathrm{1}}{\mathrm{2}\left({e}+\mathrm{1}\right)}+\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{log}^{\mathrm{7}} \left(\frac{\mathrm{e}+\mathrm{1}}{\mathrm{e}−\mathrm{1}}\right)\right]+\frac{\mathrm{26}}{{e}^{\mathrm{2}} −\mathrm{1}} \\ $$