How do you integrate # x^2/ [x^2+x+4]# using partial fractions?

Question

Evelyn Agard · Accepted Answer

#int frac{x^2}{x^2+x+4} "d"x #

#= x - 1/2 ln(x^2+x+4) + 7/sqrt15 tan^{-1}(frac{2x + 1}{sqrt15}) + c#

where #c# is the constant of integration.

First, put the partial fractions in writing. #frac{x^2}{x^2+x+4} = frac{x^2+(x+4) - (x+4)}{x^2+x+4}# #= 1 - frac{x+4}{x^2+x+4}# Fractions of the form #(f'(x))/f(x)# integrate to become #ln(f(x))#. Therefore, suppose #f(x) = x^2 + x +4#, then #f'(x) = 2x + 1# Putting the partial fraction back in writing #frac{x^2}{x^2+x+4} = 1 - 1/2 frac{2(x+4)}{x^2+x+4}# #= 1 - 1/2 frac{2x+8}{x^2+x+4}# #= 1 - 1/2 frac{2x+1}{x^2+x+4} + frac{7}{2} frac{1}{x^2+x+4}# The last bit requires completing the square, as fractions of the form #1/((x-b)^2 +a^2)# integrate to form #1/a tan^{-1}((x-b)/a)#. #x^2 + x + 4 = (x + 1/2)^2 + 15/4# Putting the partial fraction back in writing #frac{x^2}{x^2+x+4} = 1 - 1/2 frac{2x+1}{x^2+x+4} + frac{14}{(2x + 1)^2 + 15}# Currently, executing the integration #int frac{x^2}{x^2+x+4} "d"x = int "d"x - 1/2 int frac{2x+1}{x^2+x+4} "d"x # #+ 7/2 int frac{1}{(x + 1/2)^2 + 15/4} "d"x# #= x - 1/2 ln(x^2+x+4) + 7/sqrt15 tan^{-1}(frac{2x + 1}{sqrt15}) + c# where #c# is the constant of integration.

Christopher Agresta · Answer

undefined To integrate $ \frac{x^2}{x^2+x+4} $ using partial fractions, follow these steps:

1. First, factor the denominator $ x^2 + x + 4 $ if possible. In this case, it cannot be factored further over the real numbers.

2. Express $ \frac{x^2}{x^2+x+4} $ as a sum of partial fractions: $ \frac{x^2}{x^2+x+4} = \frac{A}{x - \alpha} + \frac{Bx + C}{x^2 + bx + c} $, where $ \alpha $ is a root of the denominator.

3. Multiply both sides of the equation by the denominator $ x^2 + x + 4 $ to clear the fractions.

4. Expand the resulting equation and collect like terms.

5. Equate coefficients of like terms on both sides of the equation.

6. Solve the resulting system of equations to find the values of $ A $, $ B $, and $ C $.

7. Once you have the values of $ A $, $ B $, and $ C $, rewrite the original integral using the partial fraction decomposition.

8. Now, integrate each term separately.

9. Finally, simplify the result if necessary.