∫4x+62x^2+5x+3dx =∫2(2x+3)2x^2+2x+3x+3dx =∫2(2x+3)2x(x+1)+3(x+1)dx =2∫(2x+3)(x+1)(2x+3)dx =2∫dx(x+1) =2log|x+1|+c ...

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Higher Order Derivatives

∫ 4x+6 2x 2 +...

Question

$\int \frac{4 x + 6}{{2 x}^{2} + 5 x + 3} d x$

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Solution

$\int \frac{4 x + 6}{2 x^{2} + 5 x + 3} d x$
$= \int \frac{2 (2 x + 3)}{2 x^{2} + 2 x + 3 x + 3} d x$
$= \int \frac{2 (2 x + 3)}{2 x (x + 1) + 3 (x + 1)} d x$
$= 2 \int \frac{(2 x + 3)}{(x + 1) (2 x + 3)} d x$
$= 2 \int \frac{d x}{(x + 1)}$
$= 2 log | x + 1 | + c$

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