Question

Evaluate: `∫(1 + x)/"x + e"^(- x) dx`

Answer 1

Step 1: Observe the denominator

Denominator: x + e ^−x

Differentiate it: `d/dx(x + e^(−x) = 1 − e^(−x))`

That is not exactly numerator 1 + x, so direct substitution won't work.

Step 2: Multiply the numerator and denominator by e^x

= `∫((1 + x)e^x)/(xe^x + 1) dx`

Now the denominator becomes:

xe^x + 1

Differentiate the denominator:

`d/dx(xe^x + 1) = e^x + xe^x = e^x(1 + x)`

Step 3: Substitution

u = xe^x + 1

Then

du = ex(1 + x)dx

Integral becomes:

`∫(du)/u`

Step 4: Integrate

= ln ∣u∣ + C

Substitute back:

= ln ∣xe^x + 1∣ + C