Evaluate the indefinite integral of
InT(-x^3+9x^2-3x+2)/(x^4-2x^3)
I got ((1-2x)/(2x^2))+3*ln(2-x)-4*ln(x)+c
but apparently that's not the answer... and I don't know why.
See here:
http://www.jiskha.com/display.cgi?id=1316565092#1316565092.1316570036
Thank you very much! I wasn't plugging in the absolute value signs! Thanks for the explanation too. =)
To evaluate the indefinite integral of the given function, we can start by factoring the denominator of the rational function, which is x^4 - 2x^3.
x^4 - 2x^3 = x^3(x - 2)
Now, we can rewrite the original function as:
∫ [-x^3 + 9x^2 - 3x + 2] / [x^3 (x - 2)] dx
Next, we can split this rational function into partial fractions. To do this, we need to find the partial fraction decomposition of the function.
The partial fraction decomposition takes the form:
[-x^3 + 9x^2 - 3x + 2] / [x^3 (x - 2)] = A/x + B/x^2 + C/x^3 + D/(x - 2)
To find the values of A, B, C, and D, we can multiply both sides of the equation by the denominator:
[-x^3 + 9x^2 - 3x + 2] = A(x^2)(x - 2) + B(x)(x - 2) + C(x - 2) + D(x^3)
Now, we can simplify the right-hand side of the equation:
-x^3 + 9x^2 - 3x + 2 = Ax^3 - 2Ax^2 + Bx^2 - 2Bx + Cx - 2C + Dx^3
Next, we can gather like terms:
(-1 + A + D)x^3 + (9 - 2A + B - 2B + C)x^2 + (-3 - 2B + C)x + (2 - 2C) = 0
To satisfy this equation for all x, the coefficients of corresponding powers of x must be equal.
For the coefficient of x^3, we have:
-1 + A + D = 0 ----(1)
For the coefficient of x^2, we have:
9 - 2A + B = 0 ----(2)
For the coefficient of x^1, we have:
-3 - 2B + C = 0 ----(3)
For the constant term, we have:
2 - 2C = 0 ----(4)
Now, we solve this system of equations to find the values of A, B, C, and D.
Starting with equation (1):
-1 + A + D = 0
Since we don't have any other coefficient for x^3, we can solve this equation immediately to find A:
A = 1 - D
Next, equation (4):
2 - 2C = 0
Solving for C:
C = 1
Now, substitute A = 1 - D and C = 1 into equation (3):
-3 - 2B + 1 = 0
-2B - 2 = 0
-2B = 2
B = -1
Finally, substitute A = 1 - D, B = -1, and C = 1 into equation (2):
9 - 2(1 - D) - 1 = 0
9 - 2 + 2D - 1 = 0
10 - 2 + 2D = 0
2D = -8
D = -4
Thus, we have found the values of A, B, C, and D:
A = 1 - D = 1 - (-4) = 5
B = -1
C = 1
D = -4
Now, we can express the original function as:
[-x^3 + 9x^2 - 3x + 2] / [x^3 (x - 2)] = 5/x + (-1)/x^2 + 1/x^3 + (-4)/(x - 2)
Now, we integrate each term individually:
∫(5/x) dx = 5 ln(|x|) + C1
∫(-1/x^2) dx = 1/x + C2
∫(1/x^3) dx = -1/(2x^2) + C3
∫(-4/(x - 2)) dx = -4 ln(|x - 2|) + C4
Where C1, C2, C3, and C4 are integration constants.
Therefore, the indefinite integral of the given function is:
5 ln(|x|) + 1/x - 1/(2x^2) - 4 ln(|x - 2|) + C
So, your answer was almost correct! There is just a small sign error in the integral of 1/x term, which should be positive instead of negative.