How do you integrate #f(x)=(x^2+x)/((x^2-1)(x+3)(x-9))# using partial fractions?

Question

Samantha Adkison · Accepted Answer

#-1/16*ln|x+3|+3/32ln|x-9|-1/32ln|x-1|+C# Converting your Integrand into partial fractions we get #f(x)=-1/(12*(x-1))+3/(32(x-9))-1/(16*(x+3))# Integrating this we get the result above.

Paisley Johnson · Answer

#int(x^2+x)/((x^2-1)(x+3)(x-9))dx=1/22ln(|x-1|)-5/44ln(|x+3|)+3/44ln(|x-9|)+C#, #C in RR#

#I=int(x^2+x)/((x^2-1)(x+3)(x-9))dx#

#=int(xcancel((x+1)))/(cancel((x+1))(x-1)(x+3)(x-9)dx#

Consider that #x/((x-1)(x+3)(x-9))=A/(x-1)+B/(x+3)+C/(x-9)#

#x=A(x+3)(x-9)+B(x-1)(x-9)+C(x-1)(x+3)#

#x=(A+B+C)x^2+(2C-6A-10B)x-27A+9B-3C#

By identification :

#[(A+B+C),(-6A-10B+2C),(-27A+9B-3C)]=[(0),(1),(0)]#

Solving this system, you will find :

#A=1/22#, #B=-5/44#, #C=3/44#

So:

#I=int((1/22)/(x-1)-(5/44)/(x+3)+(3/44)/(x-9))dx#

#=1/22int1/(x-1)dx-5/44int1/(x+3)dx+3/44int1/(x-9)dx#

#=1/22ln(|x-1|)-5/44ln(|x+3|)+3/44ln(|x-9|)+C#, #C in RR#

\0/ Here's our answer !

Aurora Alvarado · Answer

undefined To integrate $ f(x) = \frac{x^2 + x}{(x^2 - 1)(x + 3)(x - 9)} $ using partial fractions, follow these steps:

1. Factor the denominator completely.
2. Express $ f(x) $ as the sum of partial fractions with undetermined coefficients.
3. Clear the denominators by multiplying both sides of the equation by the common denominator.
4. Equate the numerators of the original expression and the expression obtained after clearing the denominators.
5. Solve for the undetermined coefficients.
6. Integrate each term separately.

Factoring the denominator:

$ (x^2 - 1)(x + 3)(x - 9) = (x - 1)(x + 1)(x + 3)(x - 9) $

Express $ f(x) $ as the sum of partial fractions:

$ f(x) = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{x + 3} + \frac{D}{x - 9} $

Clear the denominators:

$ x^2 + x = A(x + 1)(x + 3)(x - 9) + B(x - 1)(x + 3)(x - 9) + C(x - 1)(x + 1)(x - 9) + D(x - 1)(x + 1)(x + 3) $

Equate the numerators:

$ x^2 + x = (A + B + C + D)x^3 + (\text{terms involving } x^2) + (\text{terms involving } x) + (\text{constant term}) $

Solve for the coefficients A, B, C, and D.

After finding the values of A, B, C, and D, integrate each term separately.