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14. int(x^2+x+1)/((x+1)^2(x+2))dx...

14. `int(x^2+x+1)/((x+1)^2(x+2))`dx

Text Solution

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Given `int (x^2+x+1)/((x+1)^2(x+2))dx`
Let `(x^2+x+1)/((x+1)^2(x+2))=A/(x+2)+(Bx+C)/(x^2+1)`
`=>x^2+x+1=A(x^2+1)+(Bx+C)(x+2)`...........(1)
By solving we get `A=3/5 , B=2/5, C=1/5`
`:. I= int((3/5)/(x+2)+((2x)/5+1/5)/(x^2+1))dx`
`=3/5 int 1/(x+2)dx+1/5 int (2x)/(x^2+1)dx+1/5 int1/(x^2+1)dx`
`=3/5 log|x+2|+1/5 log|x^2+1|+1/5 tan^(-1)x+C`
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