Menu Close

Question-44498




Question Number 44498 by peter frank last updated on 30/Sep/18
Commented by maxmathsup by imad last updated on 30/Sep/18
changement 4^x =t give  e^(xln(4)) =t ⇒xln(4)=ln(t) ⇒dx=(1/(tln(4)))dt  I = ∫4^4^t   .4^t  t  (dt/(t ln(4))) =(1/(ln(4))) ∫  4^t . 4^4^t    dt   also changement  4^t =u give  dt =(du/(u ln(4))) ⇒ I = (1/(ln(4)))∫  u .4^u  (du/(u ln(4))) =(1/(ln(4))) ∫  4^u  du  =(1/(ln(4)^2 )) ∫   e^(uln(4)) du =(1/((ln(4))^3 )) 4^u    =(1/((ln(4))^3 )) 4^4^t   =(1/((ln(4))^3 )) 4^4^4^x    +c  perhaps there is something wrong in the question..!
$${changement}\:\mathrm{4}^{{x}} ={t}\:{give}\:\:{e}^{{xln}\left(\mathrm{4}\right)} ={t}\:\Rightarrow{xln}\left(\mathrm{4}\right)={ln}\left({t}\right)\:\Rightarrow{dx}=\frac{\mathrm{1}}{{tln}\left(\mathrm{4}\right)}{dt} \\ $$$${I}\:=\:\int\mathrm{4}^{\mathrm{4}^{{t}} } \:.\mathrm{4}^{{t}} \:{t}\:\:\frac{{dt}}{{t}\:{ln}\left(\mathrm{4}\right)}\:=\frac{\mathrm{1}}{{ln}\left(\mathrm{4}\right)}\:\int\:\:\mathrm{4}^{{t}} .\:\mathrm{4}^{\mathrm{4}^{{t}} } \:\:{dt}\:\:\:{also}\:{changement}\:\:\mathrm{4}^{{t}} ={u}\:{give} \\ $$$${dt}\:=\frac{{du}}{{u}\:{ln}\left(\mathrm{4}\right)}\:\Rightarrow\:{I}\:=\:\frac{\mathrm{1}}{{ln}\left(\mathrm{4}\right)}\int\:\:{u}\:.\mathrm{4}^{{u}} \:\frac{{du}}{{u}\:{ln}\left(\mathrm{4}\right)}\:=\frac{\mathrm{1}}{{ln}\left(\mathrm{4}\right)}\:\int\:\:\mathrm{4}^{{u}} \:{du} \\ $$$$=\frac{\mathrm{1}}{{ln}\left(\mathrm{4}\right)^{\mathrm{2}} }\:\int\:\:\:{e}^{{uln}\left(\mathrm{4}\right)} {du}\:=\frac{\mathrm{1}}{\left({ln}\left(\mathrm{4}\right)\right)^{\mathrm{3}} }\:\mathrm{4}^{{u}} \:\:\:=\frac{\mathrm{1}}{\left({ln}\left(\mathrm{4}\right)\right)^{\mathrm{3}} }\:\mathrm{4}^{\mathrm{4}^{{t}} } \:=\frac{\mathrm{1}}{\left({ln}\left(\mathrm{4}\right)\right)^{\mathrm{3}} }\:\mathrm{4}^{\mathrm{4}^{\mathrm{4}^{{x}} } } \:+{c} \\ $$$${perhaps}\:{there}\:{is}\:{something}\:{wrong}\:{in}\:{the}\:{question}..! \\ $$
Commented by peter frank last updated on 30/Sep/18
 true sir its typing error (/((log_(e ) 4)^3 ))
$$\:\mathrm{true}\:\mathrm{sir}\:\mathrm{its}\:\mathrm{typing}\:\mathrm{error}\:\frac{}{\left(\mathrm{log}_{\mathrm{e}\:} \mathrm{4}\right)^{\mathrm{3}} }\:\:\:\:\:\: \\ $$

Leave a Reply

Your email address will not be published.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