Integration by Substitution using Partial Fractions Decomposition

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Homework Statement

Integrate $$\int$$$$\frac{dz}{1+e^z}$$ by substitution

Homework Equations

The Attempt at a Solution

I chose u=(1+$$e^{z}$$) so du/dz=$$e^{z}$$ and dz=du/$$e^{z}$$.

Therefore, $$\int$$$$\frac{1}{u}$$ $$\frac{du}{e^{z}}$$

I plug z=ln(u-1) in for z, so $$\int$$$$\frac{1}{u}$$ $$\frac{du}{u-1}$$

From here though I don't know how to integrate. Can anyone help me with the next step?

 

[Mark44]

Rewrite 1/(u(u -1)) as a sum: A/u + B/(u - 1). Solve for A and B so that the two expressions are identically equal. This is called partial fractions decomposition.