Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 335: 87

Please see proof in "step by step"

By definition, $\displaystyle \cot u=\frac{\cos u}{\sin u}$ We note that the numerator is the derivative of the denominator, so, after substituting$ \left[\begin{array}{l} t=\sin u\\ dt=\cos udu \end{array}\right]$, the integral becomes $\displaystyle \int\frac{1}{t}dt$= Log Rule =$ \ln|t|+C=\ln|\sin u|+C$