I would like to integrate: \(\displaystyle\int{\frac{{{\left.{d}{t}\right.}}}{{{e}^{{{2}{t}}}{\left({e}^{{t}}+{1}\right)}}}}\)

Jasiah Vincent

Jasiah Vincent

Answered question

2022-04-08

I would like to integrate:
dte2t(et+1)

Answer & Explanation

Egerlandsq0z

Egerlandsq0z

Beginner2022-04-09Added 14 answers

Let x=et. Then dx=etdt
Hence, the integral becomes
dxx3(x+1)
1x3(x+1)=1x31x2+1x1x+1
Now integrate to get the answer
dxx3(x+1)=dxx3dxx2+dxxdxx+1=12x2+1x+log(x)log(x+1)
Plug in x=et to get
dte2t(et+1)=t+1et12e2tlog(et+1)

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