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`int (4x)/(x^3 + x^2 + x + 1) dx` Evaluate the integral

`int(4x)/(x^3+x^2+x+1)dx`


To solve, apply the partial fraction decomposition.  


To do so, factor the denominator.


`int(4x)/(x^3+x^2+x+1)dx = int(4x)/((x+1)(x^2+1))dx`


Then, express the integrand as sum of proper rational expressions.


`(4x)/((x+1)(x^2+1))=A/(x+1)+(Bx+C)/(x^2+1)`


Multiply both sides by the LCD.


`4x =A(x^2+1)+(Bx+C)(x+1)`


`4x = Ax^2+A + Bx^2+Bx+Cx+C`


`4x=(A+B)x^2+(B+C)x + A+C`


Express the left side as a polynomial with degree 2.


`0x^2+4x+0=(A+B)x^2+Cx+A+C`


For the two sides to be equal, the two polynomials should be the same. So set the coefficients of the two polynomials equal to each other.


x^2:


`0=A+B`     (Let this be EQ1.)


x:


`4=B+C`     (Let this be EQ2.)    


Constant:


`0=A+C`    (Let this be EQ3.)


To solve for the values of A, B and C, isolate the A in EQ1 and the C in EQ2.


EQ1:    


 `0=A+B`


`-B=A`


EQ2:  


`4 = B + C`


`4 - B = C`


Plug-in them to EQ3.


EQ3:


`0=A+C`


`0=-B+4-B`


`0=-2B+4`


`-4=-2B`


`2=B`


And, plug-in the value of B to EQ1 and EQ2.


EQ1:


`0 =A + B`


`0=A+2`


`-2=A`


EQ2:


`4=B+C`


`4=2+C`


`2=C`


So the partial fraction decomposition of the integrand is:


`(4x)/(x^3+x^2+x+1) = -2/(x+1) + (2x+2)/(x^2+1)=-2/(x+1)+(2x)/(x^2+1)+2/(x^2+1)`


Then, take the integral of it.


`int (4x)/(x^3+x^2+x+1)dx`


`=int (-2/(x+1)+ (2x)/(x^2+1) + 2/(x^2+1))dx`


`=int -2/(x+1)dx + int(2x)/(x^2+1)dx + int2/(x^2+1)dx`


`=-2ln|x+1| + ln|x^2+1|+2tan^(-1)(x) +C`



Therefore, `int (4x)/(x^3+x^2+x+1)dx=-2ln|x+1| + ln|x^2+1|+2tan^(-1)(x) +C` .         

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