`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...
`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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