When using partial fractions to find antiderivatives in calculus we decompose complicated rational expressions into the sum of simpler expressions that can be integrated individually.

Question

The required decomposition is 64/(x(x^2+8))=(A/x)+(Bx+C)/(x^2+8)

Find the values of the missing constants. A= 8 B= -8 C= ?

Answer 1

First get rid of the fractions. We can do that by multiplying each side by the LCD, which is x(x²+8)

This gives:

64 = A(x²+8) + x(Bx+C)

Next expand it out to make it a polynomial

64 = Ax² + 8A + Bx² + Cx

Next, combine like terms on each side

64 = (A+B)x² + Cx + 8A

Lastly, match the coefficients

Think of this equation as:

0x² + 0x + 64 = (A+B)x² + Cx + 8A

The coefficients in front of the x² have to be the same on both sides

Therefore, 0 = A+B

The coefficients in front of the x have to be the same on both sides

Therefore, 0 = C

Lastly, the constants at the end have to be the same on both sides

Therefore, 64 = 8A