(e^2x+1)/e^x dxintegrate by substitution method

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hala718 eNotes educator| Certified Educator

intg (e^2x+1)/e^x dx

Let t = e^x

==> dt = e^x dx = tdx ==> dx = dt/t

==> intg (e^2x+ 1)/e^x) dx = intg (t^2 + 1)/t * dt/t

                                             = intg (t^2 + 1)/t^2 dt

                                              = intg (1 + 1/t^2) dt

                                             = intg (1+ t^-2) dt

                                            = t - t^-1 + c

                                              = e^x - e^-x + c

           

neela | Student

To integrate (e^2x+1)/e^x

Solution:

(e^2x+1)/e^x = e^2x/e^x +1/e^x= e^x + e^(-x)

Int [(e^2x +1)/e^x ]dx = Int (e^x+e^-x)dx = e^x+e^(-x)/(-1) = e^x-e^-x + C, where C is the constant of integration.Substitution method:

Substitution method:

Substitute e^x = t. Then, e^x dx = dt. Or dx = dt/t.

Int [e^2x+1)/e^x]dx  =  Int{(t^2+1)/t}dt/t = Int(1+1/t^2)dt

=Int1*dt + Int(1/t^2) dt

= t + t^(-2+1)/(-2+1)

= t - t^(-1)

= e^x -e^(-x) + Constant

= t

(e^2x+1)/e^x = (t^2+1)/t