Mathematics

Integrate $$\displaystyle \int {\frac{{{v^2}}}{{{v^2} + 2v + 1}}} dx$$


SOLUTION
$$ \int \frac{v^{2}}{v^{2}+2v+1} dv $$
$$ \Rightarrow \int \frac{v^{2} + (2v+1) - (2v+1)}{v^{2} + 2v + 1} dv $$
$$ \Rightarrow \int \frac{(v^{2} + 2v + 1) - (2v+1)}{v^{2}+2v+1} dv = \int dv - \int \frac{2v+1}{v^{2}+2v+1}dv$$
$$ \Rightarrow v-\int \frac{2v+1 +(1-1)}{v^{2}+2v+1} dv $$
$$ \Rightarrow v-\int \frac{2v+2-1)}{v^{2}+2v+1} dv $$
$$ \Rightarrow v-\left [ \int \frac{2v+2dv}{v^{2}+2v+1} - \int \frac{dv}{v^{2}+2v+1} \right ]$$
$$ \Rightarrow v - \left [ \int \frac{(2v+2)dv}{v^{2}+2v+1} - \int \frac{dv}{v^{2}+2v+1} \right ]$$
Let $$ v^{2}+2v+1 = t $$
(2v+2)dv =dt 
$$ \Rightarrow v-\left [ \int \frac{dt}{t} - \int \frac{dv}{(v+1)^{2}} \right ]$$
$$ \Rightarrow v- ln\left | t \right |$$ $$ + \frac{-1}{v+1} + c $$
$$ \Rightarrow v - ln \left | v^{2}+2v+1\right |$$ - $$ \frac{1}{v+1} + c $$
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