$\int \frac{3\mathrm{x}-1}{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}\mathrm{dx}$ is equal to

$\begin{array}{l}\text{Let }\frac{3\mathrm{x}-1}{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}=\frac{\mathrm{A}}{(\mathrm{x}-1)}+\frac{\mathrm{B}}{(\mathrm{x}-2)}+\frac{\mathrm{C}}{(\mathrm{x}-3)}\\ \begin{array}{l}\Rightarrow \frac{3\mathrm{x}-1}{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}\\ =\frac{\mathrm{A}(\mathrm{x}-2)(\mathrm{x}-3)+\mathrm{B}(\mathrm{x}-1)(\mathrm{x}-3)+\mathrm{C}(\mathrm{x}-1)(\mathrm{x}-2)}{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}\\ \Rightarrow 3\mathrm{x}-1=\mathrm{A}\left[{\mathrm{x}}^2-5\mathrm{x}+6\right]+\mathrm{B}\left[{\mathrm{x}}^2-4\mathrm{x}+3\right]+\mathrm{C}\left[{\mathrm{x}}^2-3\mathrm{x}+2\right]\\ \Rightarrow 3\mathrm{x}-1={\mathrm{x}}^2(\mathrm{A}+\mathrm{B}+\mathrm{C})+\mathrm{x}(-5\mathrm{A}-4\mathrm{B}-3\mathrm{C})+(6\mathrm{A}+3\mathrm{B}+2\mathrm{C})\end{array}\end{array}$

On equating the coefficients of x²,xand constant term on both sides, we get

$\begin{array}{l}\mathrm{A}+\mathrm{B}+\mathrm{C}=0----\left(\mathrm{i}\right)\\ -5\mathrm{A}-4\mathrm{B}-3\mathrm{C}=3----\left(\mathrm{ii}\right)\\ 6\mathrm{A}+3\mathrm{B}+2\mathrm{C}=-1----\left(\mathrm{iii}\right)\end{array}$

From Eq. (i), we get A = -(B + C)
On putting the value of A in Eqs. (ii) and (iii), we get

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-5\{-(\mathrm{B}+\mathrm{C}\left)\right\}-4\mathrm{B}-3\mathrm{C}=3\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\mathrm{B}+5\mathrm{C}-4\mathrm{B}-3\mathrm{C}=3\\ \Rightarrow \mathrm{B}+2\mathrm{C}=3---\left(\mathrm{iv}\right)\\ \text{ and }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}6\{-(\mathrm{B}+\mathrm{C})+3\mathrm{B}+2\mathrm{C}=-1\\ \Rightarrow -6\mathrm{B}-6\mathrm{C}+3\mathrm{B}+2\mathrm{C}=-1\\ \Rightarrow -3\mathrm{B}-4\mathrm{C}=-1----\left(\mathrm{v}\right)\end{array}$

On solving Eqs. (iv1 and (v), we get C = 4
On putting the value of C in Eq. (iv), we get

$\begin{array}{r}\mathrm{B}+2\times 4=3\\ \Rightarrow \mathrm{B}=-5\end{array}$

On putting the value of Band C in Eq. (i), we get

$\begin{array}{l}\mathrm{A}+(-5)+4=0\Rightarrow \mathrm{A}=1\\ \therefore \mathrm{A}=1,\mathrm{B}=-5,\mathrm{C}=4\\ \text{ Now, }\int \frac{3\mathrm{x}-1}{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}\mathrm{dx}=\int \left[\frac{\mathrm{A}}{(\mathrm{x}-1)}+\frac{\mathrm{B}}{(\mathrm{x}-2)}+\frac{\mathrm{C}}{(\mathrm{x}-3)}\right]\mathrm{dx}\\ =\int \frac{1}{(\mathrm{x}-1)}\mathrm{dx}+\int \frac{(-5)}{(\mathrm{x}-2)}\mathrm{dx}+\int \frac{4}{(\mathrm{x}-3)}\mathrm{dx}\\ =\log |\mathrm{x}-1|-5\log |\mathrm{x}-2|+4\log |\mathrm{x}-3|+\mathrm{C}\\ \left[\because \frac{1}{\mathrm{x}}\mathrm{dx}=\log \mathrm{x}\right]\end{array}$