# Two ways to integrate

Back in high school, we were taught to integrate by decomposing it into partial fractions, while was to be integrated using trigonometric substitution (). It bothered me how the two methods were so vastly different just because of a change of sign. But hey, who says we can’t integrate with partial fractions?

\begin{aligned} \int \! \frac{1}{1+x^2} \mathrm{d}x &= \int \! \frac{1}{1-(ix)^2} \mathrm{d}x \\ &= \frac{1}{2} \int \! \frac{1}{1-ix} + \frac{1}{1+ix} \mathrm{d}x \\ &= \frac{1}{2} \left[\frac{1}{-i} \ln{(1-ix)} + \frac{1}{i} \ln{(1+ix)}\right] + C\\ &= \frac{1}{2} \left[i\ln{(1-ix)} - i\ln{(1+ix)}\right] + C\\ &= \frac{i}{2} \ln{\frac{1-ix}{1+ix}} + C \end{aligned}

This may look like a strange beast, but it really is How do we show this? Based on Euler’s formula, we can write as . Substitute that into the integrated expression and see for yourself!

06 January 2011