Solve- -2v2v2 - v - 2dv

I=∫2v^2v^2+v 2dv=∫2v^2+2v 4v^2+v 2dv ∫(2v 4)v^2+v 2dv =∫2dv ∫(2v+1 5)dv(v^2+v 2) I=2v ∫2v+1)v^2+v+2dv+∫5dvv^2+2v v 2 I=2v (ln|v^2+v 2|+5∫dvv ...

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Byju's Answer

Standard XII

Mathematics

Particular Solution of a Differential Equation

Solve: ∫2v2v...

Question

Solve:
$\int \frac{2 v^{2}}{v^{2} + v - 2} d v$

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Solution

$I = \int \frac{2 v^{2}}{v^{2} + v - 2} d v = \int \frac{2 v^{2} + 2 v - 4}{v^{2} + v - 2} d v - \int \frac{(2 v - 4)}{v^{2} + v - 2} d v$

$= \int 2 d v - \int \frac{(2 v + 1 - 5) d v}{(v^{2} + v - 2)}$

$I = 2 v - \int \frac{2 v + 1)}{v^{2} + v + 2} d v + \int \frac{5 d v}{v^{2} + 2 v - v - 2}$

$I = 2 v - (ln | v^{2} + v - 2 | + 5 \int \frac{d v}{v (v + 2 - (v + 2)}$

$I = 2 v - (ln | v^{2} + v - 2 | + 5 \int \frac{d v}{(v - 1) (v + 2)}$

$I = 2 v - (ln | v^{2} + v - 2 | + \frac{5}{3} \int d v [\frac{1}{(v - 1)} - \frac{1}{(v + 2)}]$

$I = 2 v + \frac{5}{3} I n | v - 1 | - ln | v - 1 | - ln | v + 2 | - \frac{5}{3} ln | v + 2 | + C$

$I = 2 v + \frac{2}{3} ln | v - 1 | - \frac{8}{3} ln | v + 2 | + C$

$I = 2 v + ln ∣ ∣ ∣ ∣ \frac{(v - 1)^{2 / 3}}{(v + 2)^{8 / 3}} ∣ ∣ ∣ ∣ + C$

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Solve:∫2v2v2+v−2dv

Solve:
$\int \frac{2 v^{2}}{v^{2} + v - 2} d v$