how do you go from dv/(v+2)(v-1) to -(1/3)/(v+2) + (1/3)/(v-1) dv?
You use partial fractions.
1/(v+2)(v-1) = 1/3 (1/(v-1) - 1/(v+2))
for partial fractions don't you do A/(v+2) + B/ (v-1) = (v+2)(v-1)?
how do you go from there?
can you show what you did step by step?
Surely this treated in your text. You want
A/(v-1) + B/(v+2) = 1/(v-1)(v+2)
A(v+2) + B(v-1) = 1
Av + 2A + Bv - B = 1
(A+B)v + (2A-B) = 0v + 1
For those polynomials to be identical, all of the coefficients must match. So, we have
A+B = 0
2A-B = 1
solve those and you get
A = 1/3
B = -1/3
Thus, the solution to the partial fraction breakdown.
To go from the expression dv/(v+2)(v-1) to the expression -(1/3)/(v+2) + (1/3)/(v-1) dv, we need to decompose the given expression into partial fractions. Here's how you can do it:
1. Start by factoring the denominator: (v+2)(v-1).
2. Next, express the given expression, dv/(v+2)(v-1), as a sum of two fractions with the same denominator, A/(v+2) + B/(v-1). Here, A and B are the constants that we need to find.
3. Multiply both sides of the equation by (v+2)(v-1) to remove the denominator from the original expression:
dv = A(v-1) + B(v+2)
4. Expand and simplify the right-hand side of the equation:
dv = (Av - A) + (Bv + 2B)
dv = (A + B)v + (2B - A)
5. Since this equation should hold for all values of v, the coefficients in front of v and the constant term on both sides must be equal. This leads to the following system of equations:
A + B = 0 (coefficient of v on the left-hand side = 0)
2B - A = 1/3 (constant term on the left-hand side = 1/3)
6. Solve this system of equations to find the values of A and B. You can use any method you prefer, such as substitution or elimination. For simplicity, let's use the substitution method.
From the first equation, A = -B. Substitute this value into the second equation:
2B - (-B) = 1/3
3B = 1/3
B = 1/9
Substituting this value back into the first equation:
A + (1/9) = 0
A = -1/9
7. Now that we have the values of A and B, we can rewrite the original expression dv/(v+2)(v-1) as a sum of partial fractions:
dv/(v+2)(v-1) = (-1/9)/(v+2) + (1/9)/(v-1)
8. Finally, simplify the fractions:
dv/(v+2)(v-1) = -(1/9)/(v+2) + (1/9)/(v-1)
Since -(1/9) can be written as -(1/3)(1/3):
dv/(v+2)(v-1) = -(1/3)/(v+2) + (1/9)/(v-1)
And since (1/9) can be written as (1/3)(1/3):
dv/(v+2)(v-1) = -(1/3)/(v+2) + (1/3)/(v-1)
So, the expression dv/(v+2)(v-1) is equivalent to -(1/3)/(v+2) + (1/3)/(v-1) dv.