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Integration by Partial Fractions

Solve: ∫4x +...

Question

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

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Solution

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

$I = \int \frac{d x (4 x + 6)}{(x + 1) (2 x + 3)} = \int \frac{d x (4 x + 4)}{(x + 1) (2 x + 3)} + 2 \int \frac{d x}{(x + 1) (2 x + 3)}$

$= \int \frac{4 d x}{(2 x + 3)} + 2 \int \frac{d x}{(x + 1) (2 x + 3)}$

$I = a | n | 2 x + 3 | = 2 [\frac{2}{(2 x + 3)} - \frac{1}{(x + 1)}]$

$I = 4 | n | 2 x + 3 | - 4 | n | 2 x + 3 | + 2 | n | x + 1 | + | n | c |$

$I = | n | c (x + 1)^{2} |$

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

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