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Question

Mathematics Question on integral

Integrate the rational function: 1x(x4−1)\frac {1}{x(x^4-1)}

Answer

1x(x4−1)\frac {1}{x(x^4-1)}

Multiplying numerator and denominator by x3 , we obtain

1x(x4−1)\frac {1}{x(x^4-1)} = x3x4(x4−1)\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 = 14∫dtt(t−1)\frac 14∫\frac {dt}{t(t-1)}

Let 1t(t−1)\frac {1}{t(t-1)} = At+B(t−1)\frac At+\frac {B}{(t-1)}

1=A(t−1)+Bt1 = A(t-1) + Bt ...(1)

Substituting t = 0 and 1 in (1), we obtain

A=−1 and B=1A = -1\ and \ B = 1

⇒ 1t(t−1)\frac {1}{t(t-1)} = −1t+1t−1\frac {-1}{t}+\frac {1}{t-1}

⇒ ∫$$\frac {1}{x(x^4-1)}dx = 14\frac 14 ∫$$\frac {-1}{t}+\frac {1}{t-1}dt

= 14[−log∣t∣+log∣t−1∣]+C\frac 14[-log|t|+log|t-1|]+C

= 14log ∣t−1t∣+C\frac 14log\ |\frac {t-1}{t}|+C

= 14log∣x4−1x4∣+C\frac 14log|\frac {x^4-1}{x^4}|+C