How do you find #int(x^2+1) / ( x^2 + 6x -3) dx# using partial fractions?

Question

Aurora Alvarado · Accepted Answer

#int (x^2+1)/(x^2+6x-3) dx#

#= x + ((18+11sqrt(3))/6)ln abs(x+3-2sqrt(3)) - ((54+11sqrt(3))/6)ln abs(x+3+2sqrt(3)) + C# #x^2+6x-3 = (x+3)^2-12# #= (x+3)^2-(2sqrt(3))^2# #= (x+3-2sqrt(3))(x+3+2sqrt(3))# #(x^2+1)/(x^2+6x-3) = (x^2+6x-3-6x+4)/(x^2+6x-3)# #=1 + (-6x+4)/(x^2+6x-3)# #=1 + A/(x+3-2sqrt(3)) + B/(x+3+2sqrt(3))# #=1 + (A(x+3+2sqrt(3)) + B(x+3-2sqrt(3)))/(x^2+6x-3)# #=1 + ((A+B)x + ((3+2sqrt(3))A + (3-2sqrt(3))B))/(x^2+6x-3)# Equating coefficients, we have: #{ (A+B = -6), ((3+2sqrt(3))A + (3-2sqrt(3))B = 4) :}# From the first equation: #B = -A-6# Substituting in the second equation: #(3+2sqrt(3))A+(3-2sqrt(3))(-A-6) = 4# Which simplifies to: #4sqrt(3)A-18+12sqrt(3) = 4# Adding #18-12sqrt(3)# to both sides we get: #4sqrt(3)A = 22+12sqrt(3)# Multiplying both sides by #sqrt(3)/12# we find: #A = (11sqrt(3))/6+3 = (18+11sqrt(3))/6# Then: #B = -A-6 = -(11sqrt(3))/6-9 = -(54+11sqrt(3))/6# So: #int (x^2+1)/(x^2+6x-3) dx# #= int 1 + ((18+11sqrt(3))/6)(1/(x+3-2sqrt(3))) - ((54+11sqrt(3))/6)(1/(x+3+2sqrt(3))) dx# #= x + ((18+11sqrt(3))/6)ln abs(x+3-2sqrt(3)) - ((54+11sqrt(3))/6)ln abs(x+3+2sqrt(3)) + C#

Evelyn Agard · Answer

undefined To find the integral of (x^2 + 1) / (x^2 + 6x - 3) dx using partial fractions, follow these steps:

1. Factor the denominator: x^2 + 6x - 3 = (x + 3)(x - 1).

2. Write the expression as a sum of partial fractions: (x^2 + 1) / (x^2 + 6x - 3) = A / (x + 3) + B / (x - 1), where A and B are constants to be determined.

3. Multiply both sides by the denominator to clear the fractions: x^2 + 1 = A(x - 1) + B(x + 3).

4. Expand and collect like terms: x^2 + 1 = Ax - A + Bx + 3B.

5. Combine like terms: x^2 + 1 = (A + B)x + (3B - A).

6. Equate coefficients of like terms: 
   For x terms: A + B = 0.
   For constant terms: 3B - A = 1.

7. Solve the system of equations for A and B.

8. Once you have found the values of A and B, rewrite the original integral as the sum of the integrals of the partial fractions.

9. Integrate each term separately.

10. Combine the results to obtain the final integral expression.