Integrate the rational function: \(\frac {1}{x(x^4-1)}\)
Solution and Explanation
\(\frac {1}{x(x^4-1)}\)
Multiplying numerator and denominator by x3 , we obtain
\(\frac {1}{x(x^4-1)}\) = \(\frac {x^3}{x^4(x^4-1)}\)
∴ \(∫\)\(\frac {1}{x(x^4-1)}dx\) = \(∫\)\(\frac {x^3}{x^4(x^4-1)}dx\)
Let x4 = t ⇒ 4x3dx = dt
∴ \(∫\)\(\frac {1}{x(x^4-1)}dx\) = \(\frac 14∫\frac {dt}{t(t-1)}\)
Let \(\frac {1}{t(t-1)}\) = \(\frac At+\frac {B}{(t-1)}\)
\(1 = A(t-1) + Bt \) ...(1)
Substituting t = 0 and 1 in (1), we obtain
\(A = -1\ and \ B = 1\)
⇒ \(\frac {1}{t(t-1)}\) = \(\frac {-1}{t}+\frac {1}{t-1}\)
⇒ \(∫\)\(\frac {1}{x(x^4-1)}dx\) = \(\frac 14\) \(∫\)\(\frac {-1}{t}+\frac {1}{t-1}dt\)
= \(\frac 14[-log|t|+log|t-1|]+C\)
= \(\frac 14log\ |\frac {t-1}{t}|+C\)
= \(\frac 14log|\frac {x^4-1}{x^4}|+C\)
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
