Looks like the integral is
![\displaystyle\int\frac{x^4-2x^2+4x+1}{x^3-x^2-x+1}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7Bx%5E4-2x%5E2%2B4x%2B1%7D%7Bx%5E3-x%5E2-x%2B1%7D%5C%2C%5Cmathrm%20dx)
First simplify the integrand by long division (or however you like):
![\dfrac{x^4-2x^2+4x+1}{x^3-x^2-x+1}=x+1+\dfrac{4x}{x^3-x^2-x+1}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%5E4-2x%5E2%2B4x%2B1%7D%7Bx%5E3-x%5E2-x%2B1%7D%3Dx%2B1%2B%5Cdfrac%7B4x%7D%7Bx%5E3-x%5E2-x%2B1%7D)
Also notice that
![x^3-x^2-x+1=x^2(x-1)-(x-1)=(x^2-1)(x-1)=(x+1)(x-1)^2](https://tex.z-dn.net/?f=x%5E3-x%5E2-x%2B1%3Dx%5E2%28x-1%29-%28x-1%29%3D%28x%5E2-1%29%28x-1%29%3D%28x%2B1%29%28x-1%29%5E2)
Split the last term into partial fractions:
![\dfrac{4x}{(x+1)(x-1)^2}=\dfrac a{x+1}+\dfrac b{x-1}+\dfrac c{(x-1)^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B4x%7D%7B%28x%2B1%29%28x-1%29%5E2%7D%3D%5Cdfrac%20a%7Bx%2B1%7D%2B%5Cdfrac%20b%7Bx-1%7D%2B%5Cdfrac%20c%7B%28x-1%29%5E2%7D)
![4x=a(x-1)^2+b(x+1)(x-1)+c(x+1)](https://tex.z-dn.net/?f=4x%3Da%28x-1%29%5E2%2Bb%28x%2B1%29%28x-1%29%2Bc%28x%2B1%29)
![4x=(a+b)x^2+(c-2a)x+a-b+c](https://tex.z-dn.net/?f=4x%3D%28a%2Bb%29x%5E2%2B%28c-2a%29x%2Ba-b%2Bc)
![\implies\begin{cases}a+b=0\\c-2a=4\\a-b+c=0\end{cases}\implies a=-1,b=1,c=2](https://tex.z-dn.net/?f=%5Cimplies%5Cbegin%7Bcases%7Da%2Bb%3D0%5C%5Cc-2a%3D4%5C%5Ca-b%2Bc%3D0%5Cend%7Bcases%7D%5Cimplies%20a%3D-1%2Cb%3D1%2Cc%3D2)
So the integral is equivalent to
![\displaystyle\int\left(x+1-\frac1{x+1}+\frac1{x-1}+\frac2{(x-1)^2}\right)\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cleft%28x%2B1-%5Cfrac1%7Bx%2B1%7D%2B%5Cfrac1%7Bx-1%7D%2B%5Cfrac2%7B%28x-1%29%5E2%7D%5Cright%29%5C%2C%5Cmathrm%20dx)
![=\displaystyle\frac{x^2}2+x-\ln|x+1|-\ln|x-1|-\frac2{x-1}+C](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac%7Bx%5E2%7D2%2Bx-%5Cln%7Cx%2B1%7C-%5Cln%7Cx-1%7C-%5Cfrac2%7Bx-1%7D%2BC)
![=\displaystyle\frac{x^2}2+x-\ln|x^2-1|-\frac2{x-1}+C](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac%7Bx%5E2%7D2%2Bx-%5Cln%7Cx%5E2-1%7C-%5Cfrac2%7Bx-1%7D%2BC)
Idkbdbdbdbdbebebebsjdjesbhejdjrnfnrrnfbfbfbdbdndndndndnd
Answer:
hi
Step-by-step explanation:
84 is divided into 7 84/7 =12
and it is added to
9 divided into 3 9/3 =3
12+3=15
Answer:
55?
Step-by-step explanation:
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