Evaluate the indefinite integral

Question

(a) Evaluate the indefinite integral ∫1/u(1+u) du Hint: 1/u(1 + u) = 1 + u − u /u(1 + u) .

(b) Using a substitution, and your answer in part (a), evaluate the indefinite integral ∫ 1/(1+ e^x) dx

Answer 1

(a) From the hint: 1/(u(1 + u)) = (1 + u – u)/(u(1 + u)) = (1 + u)/(u(1 + u)) – u/(u(1 + u))

= 1/u – 1/(1 + u)

So ∫du/(u(1 + u)) = ∫du/u – ∫du/(1 + u) = ln |u| – ln |1 + u| + C

(b) We can show that 1/(1 + e^x) = 1 – [e^x/(1 + e^x)]

(easily derived by long division, BTW)

So ∫dx/(1 + e^x) = ∫dx – ∫(e^xdx)/(1 + e^x) = x – ∫(e^xdx)/(1 + e^x)

To evaluate ∫(e^xdx)/(1 + e^x), Let u = 1 + e^x, du = e^xdx

So ∫(e^xdx)/(1 + e^x) = ∫du/u = ln (u) + C

Substituting u = 1 + e^x gives

∫(e^xdx)/(1 + e^x) = ln (1 + e^x) + C

So ∫dx/(1 + e^x) = x – ln (1 + e^x) + C

(no absolute values for the ln terms because 1 + e^x > 0 for all x)

Answer 2