Integrate the rational function: \(\frac {1}{(e^x-1) }\)
Show Hint
Put ex = t
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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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,
