Evaluate the following integrals: x-1 3x^2-4x+3 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{x}-1}{3\text{x}^2-4\text{x}+3}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\text{x}^2}{\text{x}^2+7\text{x}+10}\text{dx}$
$=\int\begin{Bmatrix}1-\frac{7\text{x}+10}{\text{x}^2+7\text{x}+10}\end{Bmatrix}\text{dx}$
$\text{I}=\text{x}-\int\frac{7\text{x}+10}{\text{x}^2+7\text{x}+10}\text{dx}+\text{c}_1\dots\text{(i})$
Let $\text{I}_1=\int\frac{7\text{x}+10}{\text{x}^2+7\text{x}+10}\text{dx}$
Let $7\text{x}+10=\lambda\frac{\text{d}}{\text{dx}}\Big(\text{x}^2+7\text{x}+10{}\Big)+\mu$
$=\lambda(2\text{x}+7)+\mu$
$7\text{x}+10=(2\lambda)\text{x}+7\lambda+\mu$
Comparing the coefficients of like powers of x,
$7=2\lambda$
$\Rightarrow\lambda=\frac72$
$7\lambda+\mu=10$
$\Rightarrow7\Big(\frac72\Big)+\mu=10$
$\mu=-\frac{29}{2}$

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