Home > Calculus > Integral by Partial Fractions

How do you integrate #int (x-5) / (x^2(x+1))# using partial fractions?

Answer 1

#6ln|x|+5/x-6ln|x+1|+C#

Decompose into partial fractions: #(x-5)/(x^2(x+1))-=(Ax+B)/x^2-6/(x+1)# giving #x-5-=(Ax+B)(x+1)-6x^2# Regrouping: #0x^2+x^1-5x^0-=(A-6)x^2+(A+B)x^1+Bx^0# Equating powers of #x#: #A=6#, #B=-5# giving the integrand as #(6x-5)/x^2-6/(x+1)# #=6/x-5/x^2-6/(x+1)# The #6# in the partial fractions comes from the cover-up rule, which avoids getting three simultaneous equations (one for each power of #x# in the identity) to solve.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

William Abdalla

To integrate (\int \frac{x-5}{x^2(x+1)} , dx) using partial fractions, first, express the integrand as the sum of two fractions with unknown constants:

[\frac{x-5}{x^2(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}]

Then, clear the fractions by multiplying both sides by the common denominator (x^2(x+1)). Afterward, equate coefficients of like terms to find the values of (A), (B), and (C). Finally, integrate each term separately.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.