Answer:
As given,
I = [tex]\int\limits {\frac{3x}{x^{2} - 2x} } \, dx[/tex]
Since the degree of the numerator is less than the degree of the denominator, we don't need to divide.
We factor the denominator as x(x-2)
The partial fraction decomposition is
[tex]\frac{3x}{x^{2} - 2x} = \frac{3x}{x(x-2)} = \frac{A}{x} + \frac{B}{x-2}\\ = \frac{A(x-2) + B(x)}{x(x-2)} \\[/tex]
we get
3x = A(x-2) + B(x)
⇒3x = x(A+B) - 2A
By comparing , we get
A+ B = 3 , -2A = 0
⇒A = 0 and B = 3- A = 3-0 = 3
∴ we get
A = 0, B = 3
Therefore, the integral become [tex]\int\limits {\frac{3x}{x^{2} - 2x} } \, dx = \int\limits {[\frac{0}{x} + \frac{3}{x-2} ] } \, dx[/tex]
