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.