Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr

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Recognize that the integral is of the form \(\int \frac{e^r}{1 + e^r} \, dr\). This suggests a substitution involving the denominator \(1 + e^r\) because its derivative appears in the numerator.

Let \(u = 1 + e^r\). Then, compute the derivative \(\frac{du}{dr} = e^r\), which means \(du = e^r \, dr\).

Rewrite the integral in terms of \(u\): since \(e^r \, dr = du\), the integral becomes \(\int \frac{1}{u} \, du\).

Integrate \(\int \frac{1}{u} \, du\), which is a standard integral resulting in \(\ln|u| + C\).

Substitute back \(u = 1 + e^r\) to express the answer in terms of the original variable \(r\), giving \(\ln|1 + e^r| + C\).

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