substitution integrals

Question

Answer 1

If u = e^x then du/dx = e^x so dx = du/e^x = du/u

So the substitution becomes:

∫-u/(u + 1/u)(du/u)

-∫1/(u + 1/u) du

-∫1/(u²/u + 1/u) du

-∫1/((u² + 1)/u) du

-∫u/(u² + 1) du

Let w = u² + 1

So dw/du = 2u or du = dw/(2u)

So the integral becomes:

-∫u/(w•(2u)) dw

-1/2∫1/w dw

-1/2ln(w) + C

-1/2ln(u² + 1) + C

-1/2ln(e^2x + 1) + C