Question:
Integrate the rational function: \(\frac {1}{(e^x-1) }\)
Integrate the rational function: \(\frac {1}{(e^x-1) }\)
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Put ex = t
Updated On: Oct 4, 2023
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Solution and Explanation
\(\frac {1}{(e^x-1) }\)
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 \(\frac {1}{t(t-1)}\) = \(\frac {A}{t}+\frac {B}{t-1}\)
\(1 = A(t-1)+Bt\) ...(1)
Substituting t = 1 and t = 0 in equation (1), we obtain
\(A = −1 \ and \ B = 1\)
∴ \(\frac {1}{t(t-1)}\) = \(\frac {-1}{t}+\frac {1}{t-1}\)
⇒ \(∫\)\(\frac {1}{t(t-1)} dt\) = \(log|\frac {t-1}{t}|+C\)
= \(log\ |\frac {e^x-1}{e^x}|+C\)
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For examples,
