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Question

Mathematics Question on integral

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

Answer

1(ex−1)\frac {1}{(e^x-1) }

Let ex = t ⇒ ex dx = dt

⇒ ∫$$\frac {1}{(e^x-1) } = ∫$$\frac {1}{t-1}.\frac {dt}{t}= ∫$$\frac {1}{t(t-1)} dt

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

1=A(t−1)+Bt1 = A(t-1)+Bt ...(1)

Substituting t = 1 and t = 0 in equation (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}{t(t-1)} dt = log∣t−1t∣+Clog|\frac {t-1}{t}|+C

= log ∣ex−1ex∣+Clog\ |\frac {e^x-1}{e^x}|+C