?Substitution Method
?FTC: ?(x = a,b) f(x) dx = F(x) | x = a,b = F(b) - F(a)

* * * * * *

?(x = 0,1) (e^(x) + 1)/(e^(x) + x) dx

Let u = e^(x) + x. Then,

du/dx = e^(x) + 1
du = (e^(x) + 1) dx [Rearrange the terms to make (e^(x) + 1) dx the subject]

Note that you can substitute the limits for u function!

u(0) = e^(0) + 0
==> 1

u(1) = e^1 + 1
==> e

So:

?(u = 1,e) ((e^(x) + 1) dx)/u
==> ?(u = 1,e) du/u
==> ln|u| | u = 1,e

Finally, by FTC:

ln(e) - ln(1)
==> 1 - 0
==> 1

I hope this helps!

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