How do you use partial fraction decomposition to decompose the fraction to integrate #(x^3+x^2+x+2)/(x^4+x^2)#?

Question

William Abdalla · Accepted Answer

See the explanation.

#x^4+x^2=x^2(x^2+1)# #A/x + B/x^2 + (Cx+D)/(x^2+1) =# #=(Ax(x^2+1) + B(x^2+1) + (Cx+D)x^2)/(x^2(x^2+1))=# #=(Ax^3+Ax+Bx^2+B+Cx^3+Dx^2)/(x^2(x^2+1))=# #=((A+C)x^3+(B+D)x^2+Ax+B)/(x^2(x^2+1))# #A+C=1# #B+D=1# #A=1# #B=2# #C=1-A=0# #D=1-B=-1# #int (x^3+x^2+x+2)/(x^4+x^2)dx=int dx/x + 2int dx/x^2 - int dx/(x^2+1) =# #=ln|x| - 2/x - arctanx + C#

Emilia Aguilar · Answer

undefined To decompose the fraction $ \frac{x^3 + x^2 + x + 2}{x^4 + x^2} $ using partial fraction decomposition, follow these steps:

1. Factor the denominator: $x^4 + x^2 = x^2(x^2 + 1)$.

2. Write the original fraction as a sum of partial fractions with unknown constants:
$$ \frac{x^3 + x^2 + x + 2}{x^4 + x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 1} $$

3. Clear the denominators by multiplying both sides of the equation by $ x^4 + x^2 $ to get rid of the fractions.

4. After clearing denominators and simplifying, equate coefficients of like terms on both sides of the equation.

5. Solve for the unknown constants $A$, $B$, $C$, and $D$.

6. Once you find the values of $A$, $B$, $C$, and $D$, rewrite the original fraction as the sum of these partial fractions.

7. Now, you can integrate each partial fraction separately.