Question

Integrate (3x^2 - 28x + 50)/(x^3 - 10x^2 + 25x) dx using partial fractions method.

Answer 1

\[ \frac{ A }{ x } + \frac{ B }{ x-5 } + \frac{ C }{ (x-5)^2 }\]

A = 2 and C = -3/5

How do I find B?

Answer 2

plug it any number of x

Answer 3

got it?

Answer 4

I am trying

Answer 5

Can you show it to me, please?

Answer 6

\[\frac{A}{x} + \frac{B}{(x-5)}+\frac{C}{(x-5)^2} = \frac{3x^2-28x+50}{x^3-10x^2+25x}\]
\[\frac{A(x-5)^2+Bx(x-5)+Cx}{x(x-5)^2} = \frac{3x^2-28x+50}{x(x-5)^2}\]
\[A(x^2 -10x+25) + B(x^2 - 5x) + Cx = 3x^2 - 28x + 50\]
\[Ax^2 - 10Ax + 25A +Bx^2 -5Bx+Cx = 3x^2 - 28x + 50\]
\[(A + B)x^2 - (10A + 5B - C)x +25A = 3x^2 - 28x + 50\]
Setting the coefficients equal gives:
\[A+B=3\]
\[10A + 5B - C = 28\]
\[25A = 50\]

This gives A = 2, plugging into the first gives B = 1,so
\[20 + 5 - c = 28\]
C = -3

So the result is:
\[\frac{2}{x} + \frac{1}{x-5} - \frac{3}{(x-5)^2}\]