#### Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.30 Question 34

$6 \log |x|-\log |x+1|-\frac{9}{x+1}+C$

Hint:

To solve this integration, we use partial fraction method

Given:

$\int \frac{5 x^{2}+20 x+6}{x^{3}+2 x^{2}+x} d x \\$

Explanation:

Let

\begin{aligned} &I=\int \frac{5 x^{2}+20 x+6}{x^{3}+2 x^{2}+x} d x \\ &I=\int \frac{5 x^{2}+20 x+6}{x(x+1)^{2}} d x \end{aligned}

\begin{aligned} &\frac{5 x^{2}+20 x+6}{x(x+1)^{2}}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}} \\ &5 x^{2}+20 x+6=A(x+1)^{2}+B(x)(x+1)+C(x) \\ &5 x^{2}+20 x+6=x^{2}(A+B)+x(2 A+B+C)+A \end{aligned}

Equating the similar terms

$5=A+B$                    (1)

$A=6$                          (2)

Equation (1)

\begin{aligned} &5=6+B \\ &B=-1 \\ &2 A+B+C=20 \\ &12-1+C=20 \\ &11+C=20 \\ &C=9 \end{aligned}

\begin{aligned} &\frac{5 x^{2}+20 x+6}{x(x+1)^{2}}=\frac{6}{x}-\frac{1}{x+1}+\frac{9}{(x+1)^{2}} \\ &I=6 \int \frac{d x}{x}-\int \frac{d x}{x+1}+9 \int \frac{d x}{(x+1)^{2}} \\ &I=6 \log |x|-\log |x+1|-\frac{9}{x+1}+C \end{aligned}