Question

Write out the form of the partial fraction decomposition of the function (as in Example 6). Do not determine the numerical values of the coefficients.

(a)\(\frac{{{x^4} + 1}}{{{x^3} + 4{x^3}}}\)

Short answer

\(\frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{{x^3}}} + \frac{{Dx + E}}{{{x^2} + 4}}\) is the final answer.

Step-by-step solution

Given Data

\(\frac{{{x^4} + 1}}{{{x^3} + 4{x^3}}}\)

\( = \frac{{{x^4} + 1}}{{{x^3}\left( {{x^2} + 4} \right)}}\)

Calculation of partial fraction.

\(\frac{{{x^4} + 1}}{{{x^3}\left( {{x^2} + 4} \right)}} = \frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{{x^3}}} + \frac{{Dx + E}}{{{x^2} + 4}}\)

Hence, \(\frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{{x^3}}} + \frac{{Dx + E}}{{{x^2} + 4}}\) is the final answer.