partialfraction x/(x^4+1)

Solution

partial fractions

\frac{x}{x ^{4} + 1}

- \frac{1}{2 2 ( x ^{2} + 2 x + 1 )} + \frac{1}{2 2 ( x ^{2} - 2 x + 1 )}

Solution steps

\frac{x}{x ^{4} + 1}

Factor

x^{4} + 1 : (x^{2} + 2 x + 1) (x^{2} - 2 x + 1)

= \frac{x}{( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 )}

Create the partial fraction template using the denominator

(x^{2} + 2 x + 1) (x^{2} - 2 x + 1)

\frac{x}{( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 )} = \frac{a _{1} x + a _{0}}{x ^{2} + 2 x + 1} + \frac{a _{3} x + a _{2}}{x ^{2} - 2 x + 1}

Multiply equation by the denominator

\frac{x ( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 )}{( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 )} = \frac{( a _{1} x + a _{0} ) ( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 )}{x ^{2} + 2 x + 1} + \frac{( a _{3} x + a _{2} ) ( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 )}{x ^{2} - 2 x + 1}

Simplify

x = (a_{1} x + a_{0}) (x^{2} - 2 x + 1) + (a_{3} x + a_{2}) (x^{2} + 2 x + 1)

Expand

x = a_{1} x^{3} - 2 a_{1} x^{2} + a_{1} x + a_{0} x^{2} - 2 a_{0} x + a_{0} + a_{3} x^{3} + 2 a_{3} x^{2} + a_{3} x + a_{2} x^{2} + 2 a_{2} x + a_{2}

Group elements according to powers of

x

1 \cdot x = x^{3} (a_{1} + a_{3}) + x^{2} (a_{0} + a_{2} + 2 a_{3} - 2 a_{1}) + x (a_{1} + 2 a_{2} + a_{3} - 2 a_{0}) + (a_{0} + a_{2})

Equate the coefficients of similar terms on both sides to create a list of equations

a_{0} + a_{2} = 0 a_{1} - 2 a_{0} + a_{3} + 2 a_{2} = 1 - 2 a_{1} + a_{0} + 2 a_{3} + a_{2} = 0 a_{1} + a_{3} = 0

Solve system of equations:

a_{0} = - \frac{1}{2 2}, a_{1} = 0, a_{2} = \frac{1}{2 2}, a_{3} = 0

a_{0} = - \frac{1}{2 2}, a_{1} = 0, a_{2} = \frac{1}{2 2}, a_{3} = 0

Plug the solutions to the partial fraction parameters to obtain the final result

\frac{0 \cdot x + ( - \frac{1}{2 2} )}{x ^{2} + 2 x + 1} + \frac{0 \cdot x + \frac{1}{2 2}}{x ^{2} - 2 x + 1}

Simplify

- \frac{1}{2 2 ( x ^{2} + 2 x + 1 )} + \frac{1}{2 2 ( x ^{2} - 2 x + 1 )}

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