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Question
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
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Solution
Let `I = int 1/ (e^x - 1) dx`
Put ex = t
⇒ ex dx = dt
⇒ `dx = dt/t`
∴ `I = int dt/ (t (t - 1))`
Let `1/ (t (t - 1)) = A/t + B/ (t - 1)`
⇒ 1 = A (t - 1) + Bt .....(i)
Putting t = 1 in (i), we get
B = 1
Putting t = 0 in (i), we get
1 = A (0 - 1) + B (0)
⇒ A = -1
∴ `1/ (t (t - 1)) = (-1)/t + 1/ (t - 1)`
∴ `I = int (-1/t + 1/ (t - 1)) dt`
= - log |t| + log |t - 1| + C
= - log |ex| + log |ex - 1| + C
`= log ((e^x - 1)/e^x) +C`
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