$\int \frac{d x}{x^3+3 x^2+2 x}=$

$\int \frac{d x}{x^3+3 x^2+2 x}=$
  1. $\log |x|+\log \left|\frac{x+2}{x+1}\right|+c$
  2. $\log |x|-\log |x+1|+\log |x+2|+c$
  3. $\frac{1}{2}[\log |x|+\log |x+1|+\log |x+2|]+c$
  4. $\frac{1}{2} \log \left(\frac{\left|x^2+2 x\right|}{(x+1)^2}\right)+c$

Solution

Let $\quad \begin{aligned} I & =\int \frac{d x}{x^3+3 x^2+2 x} \\ & =\int \frac{d x}{x\left(x^2+3 x+2\right)} \\ & =\int \frac{d x}{x(x+1)(x+2)}\end{aligned}$ $ \begin{aligned} & \text { Let } \frac{1}{x(x+1)(x+2)}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{x+2} \\ & \Rightarrow 1=A(x+1)(x+2)+B x(x+2)+C x(x+1) \end{aligned} $ Put $x=0$, we get $ A=\frac{1}{2} $ Put $x=-1$, we get $ B=-1 $ Put $x=-2$, we get $ \begin{aligned} & \quad C=\frac{1}{2} \\ & \therefore \quad I=\int\left(\frac{1}{2 x}-\frac{1}{x+1}+\frac{1}{2(x+2)}\right) d x \\ & =\frac{1}{2} \log x-\log (x+1)+\frac{1}{2} \log (x+2)+C \\ & =\frac{1}{2}[\log x-2 \log (x+1)+\log (x+2)]+C \\ & =\frac{1}{2} \log \left|\frac{x(x+2)}{(x+1)^2}\right|+C \\ & =\frac{1}{2} \log \left|\frac{x^2+2 x}{(x+1)^2}\right|+C . \end{aligned} $

Asked in: AP EAMCET 2018 (22 Apr Shift 1)

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