**Question**

**Introduction**

The
book

*VisualComplex Analysis by Tristan Needham*recounts the story of RogerCotes who considered the above problem. Without ostensibly using complex numbers, Cotes discovered a geometrical principle that helped to factorise the denominator*x*– 1, and hence decompose the above integral.^{n}
In this
article, I am going to “cheat” by using complex numbers to split up the denominator.
The fact that the denominator splits
completely into a product of simple linear factors makes it easy to decompose
the integrand into partial fractions. Once
this is done, I can single out the one or two fractions with purely real linear
denominators. Then I can pair up the
conjugate fractions to get fractions with real quadratic denominators. In other words, I apply a

*divide-and-conquer strategy*, splitting up a big problem into smaller problems (Heuristics!). Then I collect all the partial answers together to form my final answer.**Solution**

**Remarks**

Note that I
have only used real integration, not complex integration. Complex numbers are used only to derive the
various algebraic fractions.

H02. Use a diagram / model (mentally: imagine roots of unity in a
circle)

H04. Look for pattern(s)

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a
Mathematical Sentence**Suitable Levels**

*****University / college level calculus

*****high school students very strong and interested in calculus and complex numbers

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