Integrate -v2-v2 - 2v - 1 dx

∫v^2/v^2+2v+1 dv ⇒∫v^2 + (2v+1) (2v+1)/v^2 + 2v + 1 dv ⇒∫(v^2 + 2v + 1) (2v+1)/v^2+2v+1 dv = ∫ dv ∫2v+1/v^2+2v+1dv ⇒ v ∫2v+1 +(1 1)/v^ ...

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Byju's Answer

Standard XII

Mathematics

Particular Solution of a Differential Equation

Integrate ∫...

Question

Integrate $\int \frac{v^{2}}{v^{2} + 2 v + 1} d x$

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Solution

$\int \frac{v^{2}}{v^{2} + 2 v + 1} d v$
$\Rightarrow \int \frac{v^{2} + (2 v + 1) - (2 v + 1)}{v^{2} + 2 v + 1} d v$
$\Rightarrow \int \frac{(v^{2} + 2 v + 1) - (2 v + 1)}{v^{2} + 2 v + 1} d v = \int d v - \int \frac{2 v + 1}{v^{2} + 2 v + 1} d v$
$\Rightarrow v - \int \frac{2 v + 1 + (1 - 1)}{v^{2} + 2 v + 1} d v$
$\Rightarrow v - \int \frac{2 v + 2 - 1)}{v^{2} + 2 v + 1} d v$
$\Rightarrow v - [\int \frac{2 v + 2 d v}{v^{2} + 2 v + 1} - \int \frac{d v}{v^{2} + 2 v + 1}]$
$\Rightarrow v - [\int \frac{(2 v + 2) d v}{v^{2} + 2 v + 1} - \int \frac{d v}{v^{2} + 2 v + 1}]$
Let $v^{2} + 2 v + 1 = t$
(2v+2)dv =dt
$\Rightarrow v - [\int \frac{d t}{t} - \int \frac{d v}{(v + 1)^{2}}]$
$\Rightarrow v - l n | t |$ $+ \frac{- 1}{v + 1} + c$
$\Rightarrow v - l n ∣ ∣ v^{2} + 2 v + 1 ∣ ∣$ - $\frac{1}{v + 1} + c$

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Integrate ∫v2v2+2v+1dx

Integrate $\int \frac{v^{2}}{v^{2} + 2 v + 1} d x$