Integrate- - v 1 - v -

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Standard XII

Mathematics

Functions

Integrate: ∫...

Question

Integrate:
$\int \frac{v}{1 - v} =$

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Solution

@page { margin: 2cm } p { margin-bottom: 0.25cm; line-height: 120% }
$\begin{matrix} \Rightarrow \int \frac{1 + v - 1}{(1 - v)} \Rightarrow \int \frac{d v}{(1 - v)} - \int \frac{(1 - v) d v}{(1 - v)} \Rightarrow - log | 1 - v | - \int d v \Rightarrow log {(1 - v)}^{- 1} - v + C \Rightarrow log \frac{1}{(1 - v)} - v + C \end{matrix}$

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Integrate:∫v1−v=

@page { margin: 2cm } p { margin-bottom: 0.25cm; line-height: 120% } ⇒∫1+v−1(1−v)⇒∫dv(1−v)−∫(1−v)dv(1−v)⇒−log|1−v|−∫dv⇒log(1−v)−1−v+C⇒log1(1−v)−v+C

Integrate:
$\int \frac{v}{1 - v} =$