Question

Integrate the function: \(\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}\)

Answer 1

\(Let \space \frac{x^{2+}x+1}{(x+1)^{2}(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+1)^{2}}+\frac{C}{(x+2)}...(1)\)

\(⇒x^{2}+x+1=A(x+1)(x+2)+B(x+2)+C(x^{2}+2x+1)\)

\(⇒x^{2}+x+1=A(x^{2}+3x+2)+B(x+2)+C(x^{2}+2x+1)\)

\(⇒x^{2}+x+1=(A+C)x^{2}+(3A+B+2C)x+(2A+2B+C)\)

Equating the coefficients of \(x2,x,\) and constant term,we obtain

\(A+C=1\)

\(3A+B+2C=1\)

\(2A+2B+C=1\)

On solving these equations,we obtain

\(A=-2,B=1,and C=3\)

From equation(1),we obtain

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

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

\(=-2log|x+1|+3log|x+2|-\frac{1}{(x+1)}+C\)