#10; #10; ∫(v/1 v^n)>dv =∫(v^n 1/v^n(1 v^n))>dv Let> #10;v^n=t then>nv^n 1dv=dt or>v^n 1dv=(1/n)dt ∴>I=(1/n)∫(1 ...

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Functions

∫ v / 1 - v ...

Question

$\int \frac{v}{1 - v^{n}}$ =

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Solution

$\int (\frac{v}{1 - v^{n}}) d v = \int (\frac{v^{n - 1}}{v^{n} (1 - v^{n})}) d v L e t v^{n} = t t h e n n v^{n - 1} d v = d t o r v^{n - 1} d v = (\frac{1}{n}) d t ∴ I = (\frac{1}{n}) \int (\frac{1}{t (1 - t)}) d t = (\frac{1}{n}) \int ((\frac{1}{1 - t}) + (\frac{1}{t})) d t = (\frac{1}{n}) (- l o g | 1 - t | + l o g | t |) + c = (\frac{1}{n}) l o g ∣ ∣ (\frac{t}{1 - y}) ∣ ∣ + c = (\frac{1}{n}) l o g ∣ ∣ (\frac{v^{n}}{1 - v^{n}}) ∣ ∣ + c$

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