Evaluate the following integrals: x x^2+3x+2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\text{x}}{\text{x}^2+3\text{x}+2}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{\text{x}}{\text{x}^2+3\text{x}+2}\text{ dx}$
Let $\text{x}=\lambda\frac{\text{d}}{\text{dx}}\big(\text{x}^2+3\text{x}+2\big)+\mu$
$=\lambda(2\text{x}+3)+\mu$
$\text{x}=(2\lambda)\text{x}+(3\lambda+\mu)$
Comparing the coefficients of like powers of x,
$2\lambda=1\Rightarrow\lambda=\frac{1}{2}$
$3\lambda+\mu=0\Rightarrow3\Big(\frac{1}{2}\Big)+\mu=0$
$\mu=-\frac{3}{2}$
So, $\text{I}=\int\frac{\frac{1}{2}(2\text{x}+3)-\frac{3}{2}}{\text{x}^2+3\text{x}+2}\text{ dx}$
$\text{I}=\frac{1}{2}\int\frac{2\text{x}+3}{\text{x}^2+3\text{x}+2}\text{ dx}-\frac{3}{2}\int\frac{1}{\text{x}^2+3\text{x}+2}\text{ dx}$
$\text{I}=\frac{1}{2}\int\frac{2\text{x}+3}{\text{x}^2+3\text{x}+2}\text{ dx}-\frac{3}{2}\int\frac{1}{\text{x}^2+2\text{x}\big(\frac{3}{2}\big)+\big(\frac{3}{2}\big)^2-\big(\frac{3}{2}\big)^2+2}\text{ dx}$
$\text{I}=\frac{1}{2}\int\frac{2\text{x}+3}{\text{x}^2+3\text{x}+2}\text{ dx}-\frac{3}{2}\int\frac{1}{\big(\text{x}+\frac{3}{2}\big)^2-\big(\frac{1}{2}\big)^2}\text{ dx}$
$\text{I}=\frac{1}{2}\log\big|\text{x}^2+3\text{x}+2\big|-\frac{3}{2}\times\frac{1}{2\big(\frac{1}{2}\big)}\log\Bigg|\frac{\text{x}+\frac{3}{2}-\frac{1}{2}}{\text{x}+\frac{3}{2}+\frac{1}{2}}\Bigg|+\text{C}$ $\Big[\text{Since }\int\frac{1}{\text{a}^2-\text{x}^2}\text{ dx}=\frac{1}{2\text{a}}\log\Big|\frac{\text{x}-\text{a}}{\text{x}+\text{a}}\Big|+\text{C}\Big]$
$\text{I}=\frac{1}{2}\log\big|\text{x}^2+3\text{x}+2\big|-\frac{3}{2}\log\Big|\frac{\text{x}+1}{\text{x}+2}\Big|+\text{C}$

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