SOLUTION: Please help. Integrate {(1-cotx)^2

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Question 886474: Please help. Integrate {(1-cotx)^2
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
int%28%281-cot%28x%29%29%5E2%2Cdx%29

                          Square the binomial:

int%28%281-2cot%28x%29%2Bcot%5E2x%29%2Cdx%29

                          Write as the sum of three integrals:

int%28%281%29%2Adx%29%22%22%2B%22%22int%28%28-2cot%28x%29%29%2Adx%29%22%22%2B%22%22int%28%28cot%5E2x%29%2Adx%29

                          Take out constant in second integral, and use the
                          Pythagorean identity cot%5E2x=csc%5E2x-1 in the third integral: 


int%28dx%29%22%22-%22%222%2Aint%28%28cot%28x%29%29%2Adx%29%22%22%2B%22%22int%28%28csc%5E2x-1%29%2Adx%29

int%28dx%29%22%22-%22%222%2Aint%28%28cot%28x%29%29%2Adx%29%22%22%2B%22%22int%28%28csc%5E2x%29%2Adx%29%22%22%2B%22%22int%28%28-1%29%2Adx%29

int%28dx%29%22%22-%22%222%2Aint%28%28cot%28x%29%29%2Adx%29%22%22%2B%22%22int%28%28csc%5E2x%29%2Adx%29%22%22-%22%22int%28dx%29

                          The first and last integrals cancel

-2%2Aint%28%28cot%28x%29%29%2Adx%29%22%22%2B%22%22int%28%28csc%5E2x%29%2Adx%29

                          Use the integral formulas:

                          int%28%28cot%28u%29%29%2Cdu%29=ln%28abs+%28sin%28u%29%29+%29%2BC and int%28%28csc%5E2u%29%2Cdu%29=-cot%28u%29%2BC

-2%2Aln%28abs+%28sin%28x%29%29+%29+%2B+%28-cot%28x%29%29%2BC

-2%2Aln%28abs+%28sin%28x%29%29+%29+-+cot%28x%29%2BC

Edwin