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e-4t-e-2t-3e-t-2-dt-




Question Number 138403 by tugu last updated on 13/Apr/21
∫(e^(4t) /(e^(2t) +3e^t +2))dt=?
$$\int\frac{{e}^{\mathrm{4}{t}} }{{e}^{\mathrm{2}{t}} +\mathrm{3}{e}^{{t}} +\mathrm{2}}{dt}=? \\ $$
Answered by bemath last updated on 13/Apr/21
let e^t  = u , e^(2t) +3e^t +2=u^2 +3u+2  =(u+1)(u+2)  e^t  dt = du   ⇒I=∫(u^3 /(u^2 +3u+2)) du
$${let}\:{e}^{{t}} \:=\:{u}\:,\:{e}^{\mathrm{2}{t}} +\mathrm{3}{e}^{{t}} +\mathrm{2}={u}^{\mathrm{2}} +\mathrm{3}{u}+\mathrm{2} \\ $$$$=\left({u}+\mathrm{1}\right)\left({u}+\mathrm{2}\right) \\ $$$${e}^{{t}} \:{dt}\:=\:{du}\: \\ $$$$\Rightarrow{I}=\int\frac{{u}^{\mathrm{3}} }{{u}^{\mathrm{2}} +\mathrm{3}{u}+\mathrm{2}}\:{du} \\ $$

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