∫3x 1(x+2)^2 dx=∫3x+6 5(x+2)^2dx=∫3x+6(x+2)^2dx 5∫dx(x+2)^2=∫3x+2 5∫dx(x+2)^2 =3log(x+2) 5(x+2)^ 2+1 2+1=3log(x+2) 5(x+2)^ 1 1 =3log(x+2)+5x+2 ...

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Integration by Partial Fractions

∫3x -1x+22 dx...

Question

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

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Solution

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

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