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115. $\frac{2 x}{x^{2} + 3 x + 2}$

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Vishal Baghel | Contributor-Level 10

4 months ago

$\frac{2 x}{x^{2} + 3 x + 2} = \frac{2 x}{(x + 1) (x + 2)} = \frac{A}{(x + 1)} + \frac{B}{(x + 2)}$

2x = A(x + 2) + B(x + 1).

= (A + B) x + (2A + B).

Comparing the coefficients we get,

A + B = 2 ---(1)

2A + B = 0 ---- (2).

So, Eqⁿ (2) - (1),

2A + B - (A + B) = 0 - 2

Þ A = - 2

And B = 2 - A = 2 - (-2) = 2 + 2 = 4

B = 4

$\begin{array}{l} S o, \frac{2 x}{(x + 1) (x + 2)} = \frac{- 2}{x + 1} + \frac{4}{x + 2} . \\ ∴ \int \frac{2 x d x}{(x + 1) (x + 2)} = - 2 \int \frac{d x}{x + 1} + 4 \int \frac{d x}{x + 2} \\ = - 2 l o g | x + 1 | + 4 l o g | x + 2 | + c \\ = 4 l o g | x + 2 | - 2 l o g | x + 1 | + c \end{array}$

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