Solve the integral: (2x-3)/(x^3-x^2)
I'm stuck at finding A. I did partial fractions. I've already found B and C.
At x=1, C=-1
At x=0, B=3
Wolfram split it up into these 3 fractions:
http://www.wolframalpha.com/input/?i=partial+fractions+(2x-3)%2F(x%5E3-x%5E2)
notice, possible denominators could have been,
x, x^2, and (x-1)
How did you do your original split-up?
I did that.
∫A/x +∫B/x^2 +∫C/(x-1)
F(X)=A ln|x| + B/x + C ln|x-1| + C
(2x-3)/[x^2(x-1)] = [A(x^2)(x-1)]/[x] + [B(x^2)(x-1)]/[x^2] + [C(x^2)(x-1)]/[x-1]
2x-3 = A(x)(x-1) + B(x-1) + C(x^2)
To solve the integral (2x-3)/(x³-x²), you correctly started by applying partial fractions. Let's review the steps for finding the values of A, B, and C.
1. Start with the expression (2x-3)/(x³-x²).
2. Factor the denominator, x³-x², as x²(x-1).
3. Decompose the expression using the partial fractions method:
(2x-3)/(x³-x²) = A/x + B/x² + C/(x-1).
Now, let's find the values of A, B, and C.
4. Combine the fractions on the right side into a common denominator:
(2x-3)/(x³-x²) = A(x-1) + B(x-1)x + Cx².
Simplify further:
2x-3 = A(x-1) + B(x-1)x + Cx².
5. To solve for A, B, and C, you can either use the method of comparing coefficients or equate the numerators directly. Let's equate the numerators here.
For the x² term:
Coefficient of x² on the left: 0
Coefficient of x² on the right: C
Therefore, C = 0.
For the x term:
Coefficient of x on the left: 2
Coefficient of x on the right: A + B
Therefore, A + B = 2. (Equation 1)
For the constant term:
Constant term on the left: -3
Constant term on the right: -A
Therefore, -A = -3.
Solving for A, we get A = 3.
Using A = 3 in Equation 1,
3 + B = 2
B = 2 - 3
B = -1.
Therefore, A = 3, B = -1, and C = 0.
Now that you have found the values of A, B, and C, you can rewrite the integral as:
∫(2x-3)/(x³-x²) dx = ∫(3/x) dx - ∫(1/x²) dx.
Integrating each term separately, you get:
∫(3/x) dx = 3ln|x| + K1,
∫(1/x²) dx = -1/x + K2,
where K1 and K2 are constants of integration.
Thus, the final solution to the integral is:
∫(2x-3)/(x³-x²) dx = 3ln|x| - 1/x + K,
where K = K1 + K2 is the constant of integration.