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24 Nov 2011 12:24:55 IST

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Integration of arcCot(x^2+x+1)

Mathematics

Integration of arcCot(x^2+x+1)

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Pranjal Tomar

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24 Nov 2011 22:28:47 IST

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http://www3.wolframalpha.com/Calculate/MSP/MSP123819i5b97ce80f23ge00004b46677ei69d9467?MSPStoreType=image/gif&s=42&w=474&h=1243

saurav

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18 Dec 2011 23:35:57 IST

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Arccot(x^2+x+1)=arctan(x+1)-arctan(x).rest is obvious.

Bipin Dubey

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29 Jan 2012 18:20:01 IST

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$cot^{-1}(x^2+x+1)=cot^{-1}\left (\frac{x(x+1)+1)}{x+1-x} \right )=cot^{-1}(x+1)-cot^{-1}x$

Now use by parts for arccot :

$\int 1.cot^{-1}dx=cot^{-1}x\int 1dx-\int \left (\int 1dx \right )\left ( \frac{d}{dx}cot^{-1}x \right )dx$

$=xcot^{-1}x-\int \frac{-x}{1+x^2}dx=xcot^{-1}x+\frac{1}{2}log(x^2+1)$

$Similarly \; \int cot^{-1}(x+1)dx=(x+1)cot^{-1}(x+1)+\frac{1}{2}log((x+1)^2+1)$

Subtract these two to get the final answer

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