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Integration by Partial Fractions

Resolve 3x2...

Question

Resolve $\frac{3 x^{2} - 3 x - 11}{(x + 3) (3 x + 4)^{2}}$ into partial fractions.

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Solution

b'
Let $dfrac3x2−3x−11(x+3)(3x+4)2=dfracAx+3+dfracB3x+4+dfracC(3x+4)2$

Consider
$3 x^{2} - 3 x - 11 = A (3 x + 4)^{2} + B (x + 3) (3 x + 4) + C (x + 3)$
$= A (9 x^{2} + 16 + 24 x) + B (3 x^{2} + 13 x + 12) + C (x + 3)$
On comparison,
$9 A + 3 B = 3$
$24 A + 13 B + C = - 3$
$16 A + 12 B + 3 C = - 11$
Solving these equations, we get,
$A = 1, B = - 2, C = - 1$
Therefore,
$dfrac3x2−3x−11(x+3)(3x+4)2=dfrac1x+3−dfrac23x+4−dfrac1(3x+4)2$
'

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