Integration with $e$












1












$begingroup$


What are the steps in this integration :



$$
int4e^{t(u-4)}dt = frac{-4}{u-4}e^{t(u-4)}
$$



I've calculated simpler integration formulas such as
$$
int x^2dt = frac{x^3}{3}
$$



but I'm unsure how integral above is calculated as there is an $e$ involved.






calculus integration derivatives







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    As @Jose's answer shows, the formula you've got on the right-hand side is incorrect --- there's an extra minus-sign. So we can't show you the steps that lead to an incorrect answer.
    $endgroup$
    – John Hughes
    Dec 16 '18 at 15:42
















1












$begingroup$


What are the steps in this integration :



$$
int4e^{t(u-4)}dt = frac{-4}{u-4}e^{t(u-4)}
$$



I've calculated simpler integration formulas such as
$$
int x^2dt = frac{x^3}{3}
$$



but I'm unsure how integral above is calculated as there is an $e$ involved.






calculus integration derivatives







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    As @Jose's answer shows, the formula you've got on the right-hand side is incorrect --- there's an extra minus-sign. So we can't show you the steps that lead to an incorrect answer.
    $endgroup$
    – John Hughes
    Dec 16 '18 at 15:42














1












1








1





$begingroup$


What are the steps in this integration :



$$
int4e^{t(u-4)}dt = frac{-4}{u-4}e^{t(u-4)}
$$



I've calculated simpler integration formulas such as
$$
int x^2dt = frac{x^3}{3}
$$



but I'm unsure how integral above is calculated as there is an $e$ involved.






calculus integration derivatives







share|cite|improve this question









$endgroup$




What are the steps in this integration :



$$
int4e^{t(u-4)}dt = frac{-4}{u-4}e^{t(u-4)}
$$



I've calculated simpler integration formulas such as
$$
int x^2dt = frac{x^3}{3}
$$



but I'm unsure how integral above is calculated as there is an $e$ involved.






calculus integration derivatives




calculus integration derivatives






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 16 '18 at 15:37









blue-skyblue-sky

1,03311322




1,03311322








  • 3




    $begingroup$
    As @Jose's answer shows, the formula you've got on the right-hand side is incorrect --- there's an extra minus-sign. So we can't show you the steps that lead to an incorrect answer.
    $endgroup$
    – John Hughes
    Dec 16 '18 at 15:42














  • 3




    $begingroup$
    As @Jose's answer shows, the formula you've got on the right-hand side is incorrect --- there's an extra minus-sign. So we can't show you the steps that lead to an incorrect answer.
    $endgroup$
    – John Hughes
    Dec 16 '18 at 15:42








3




3




$begingroup$
As @Jose's answer shows, the formula you've got on the right-hand side is incorrect --- there's an extra minus-sign. So we can't show you the steps that lead to an incorrect answer.
$endgroup$
– John Hughes
Dec 16 '18 at 15:42




$begingroup$
As @Jose's answer shows, the formula you've got on the right-hand side is incorrect --- there's an extra minus-sign. So we can't show you the steps that lead to an incorrect answer.
$endgroup$
– John Hughes
Dec 16 '18 at 15:42










3 Answers
3






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2












$begingroup$

$$I=int4e^{(u-4)t}dt=4int e^{(u-4)t}dt$$
Substitute $$w=(u-4)t\Rightarrow dw=(u-4)dt\Rightarrow frac{dw}{u-4}=dt$$
Hence $$I=4int e^{w}frac{dw}{u-4}=frac{4}{u-4}int e^wdw$$
Since $frac{d}{dx}e^x=e^x$,
$$I=frac{4}{u-4}e^w+C$$
Re-substitute:
$$I=frac{4}{u-4}e^{(u-4)t}+C$$






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    Since $int e^t,mathrm dt=e^t$, $int e^{at},mathrm dt=dfrac{e^{at}}a$.






    share|cite|improve this answer









    $endgroup$





















      -1












      $begingroup$

      Substituting $$m=t(u-4)$$ then you will get $$dm=(u-4)dt$$






      share|cite|improve this answer









      $endgroup$













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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        2












        $begingroup$

        $$I=int4e^{(u-4)t}dt=4int e^{(u-4)t}dt$$
        Substitute $$w=(u-4)t\Rightarrow dw=(u-4)dt\Rightarrow frac{dw}{u-4}=dt$$
        Hence $$I=4int e^{w}frac{dw}{u-4}=frac{4}{u-4}int e^wdw$$
        Since $frac{d}{dx}e^x=e^x$,
        $$I=frac{4}{u-4}e^w+C$$
        Re-substitute:
        $$I=frac{4}{u-4}e^{(u-4)t}+C$$






        share|cite|improve this answer









        $endgroup$


















          2












          $begingroup$

          $$I=int4e^{(u-4)t}dt=4int e^{(u-4)t}dt$$
          Substitute $$w=(u-4)t\Rightarrow dw=(u-4)dt\Rightarrow frac{dw}{u-4}=dt$$
          Hence $$I=4int e^{w}frac{dw}{u-4}=frac{4}{u-4}int e^wdw$$
          Since $frac{d}{dx}e^x=e^x$,
          $$I=frac{4}{u-4}e^w+C$$
          Re-substitute:
          $$I=frac{4}{u-4}e^{(u-4)t}+C$$






          share|cite|improve this answer









          $endgroup$
















            2












            2








            2





            $begingroup$

            $$I=int4e^{(u-4)t}dt=4int e^{(u-4)t}dt$$
            Substitute $$w=(u-4)t\Rightarrow dw=(u-4)dt\Rightarrow frac{dw}{u-4}=dt$$
            Hence $$I=4int e^{w}frac{dw}{u-4}=frac{4}{u-4}int e^wdw$$
            Since $frac{d}{dx}e^x=e^x$,
            $$I=frac{4}{u-4}e^w+C$$
            Re-substitute:
            $$I=frac{4}{u-4}e^{(u-4)t}+C$$






            share|cite|improve this answer









            $endgroup$



            $$I=int4e^{(u-4)t}dt=4int e^{(u-4)t}dt$$
            Substitute $$w=(u-4)t\Rightarrow dw=(u-4)dt\Rightarrow frac{dw}{u-4}=dt$$
            Hence $$I=4int e^{w}frac{dw}{u-4}=frac{4}{u-4}int e^wdw$$
            Since $frac{d}{dx}e^x=e^x$,
            $$I=frac{4}{u-4}e^w+C$$
            Re-substitute:
            $$I=frac{4}{u-4}e^{(u-4)t}+C$$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 16 '18 at 20:04









            clathratusclathratus

            4,8611338




            4,8611338























                1












                $begingroup$

                Since $int e^t,mathrm dt=e^t$, $int e^{at},mathrm dt=dfrac{e^{at}}a$.






                share|cite|improve this answer









                $endgroup$


















                  1












                  $begingroup$

                  Since $int e^t,mathrm dt=e^t$, $int e^{at},mathrm dt=dfrac{e^{at}}a$.






                  share|cite|improve this answer









                  $endgroup$
















                    1












                    1








                    1





                    $begingroup$

                    Since $int e^t,mathrm dt=e^t$, $int e^{at},mathrm dt=dfrac{e^{at}}a$.






                    share|cite|improve this answer









                    $endgroup$



                    Since $int e^t,mathrm dt=e^t$, $int e^{at},mathrm dt=dfrac{e^{at}}a$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 16 '18 at 15:40









                    José Carlos SantosJosé Carlos Santos

                    166k22132235




                    166k22132235























                        -1












                        $begingroup$

                        Substituting $$m=t(u-4)$$ then you will get $$dm=(u-4)dt$$






                        share|cite|improve this answer









                        $endgroup$


















                          -1












                          $begingroup$

                          Substituting $$m=t(u-4)$$ then you will get $$dm=(u-4)dt$$






                          share|cite|improve this answer









                          $endgroup$
















                            -1












                            -1








                            -1





                            $begingroup$

                            Substituting $$m=t(u-4)$$ then you will get $$dm=(u-4)dt$$






                            share|cite|improve this answer









                            $endgroup$



                            Substituting $$m=t(u-4)$$ then you will get $$dm=(u-4)dt$$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Dec 16 '18 at 15:41









                            Dr. Sonnhard GraubnerDr. Sonnhard Graubner

                            77.2k42866




                            77.2k42866






























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