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Integrate the function $\int\large\frac{x^2+x+1}{(x+1)^2(x+2)}$

1 Answer

  • If the integral function is a proper rational function, then it can be resolved into its partial fraction,\[(ie)\frac{1}{(x+a)(x+b)^2}=\frac{A}{(x+a)}+\frac{B}{(x+b)}+\frac{c}{(x+b)^2}\]
Step 1:
On comparing the coefficients of like powers on both sides we get,
Step 2:
On solving these equations we get,
Substituting this in equ(4) we get
Substituting the value of $C$ in equ(5) we get,
Substituting the value of $C$ in equ(6) we get,
Step 3:
From equ(1) we get
$\qquad\qquad\;\;\;=-2\int\large\frac{dx}{x+1}$$+\int \large\frac{dx}{(x+1)^2}$$+3\int \large\frac{dx}{(x+2)}$
$\qquad\qquad\;\;\;=-2\log \mid x+1\mid-\large\frac{1}{(x+1)}$$+3\log\mid x+2\mid +c$
answered Sep 10, 2013 by sreemathi.v