How do I decompose and solve this integral?
∫((8x-5)/(x^3 -1))dx
(8x-5)/(x^3-1) = (8x-5)/((x-1)(x^2+x+1))
= A/(x-1) + (Bx+C)/(x^2+x+1)
follow the normal partial fraction solution to find that
∫((8x-5)/(x^3 -1))dx = ∫(1/(x-1) - (x-6)/(x^2+x+1) dx
Now, if u = x^2+x+1, du=2x+1 dx
x-6 = 1/2 (2x-1) - 11/2
So now we have to find
∫ 1/(x-1) dx = ln(x-1)
∫ 1/2 (du/u) = 1/2 ln(x^2+2x+1)
Now, just one step to go
x^2+x+1 = (x + 1/2)^2 + 3/4
You know that ∫1/(a^2+u^2) = arctan(u/a)
So the final result is
ln(1-x) - 1/2 ln(x^2+x+1) + 13/√3 arctan (2x+1)/√3 + C
Thank oobleck
To decompose and solve the integral ∫((8x-5)/(x^3 -1))dx, we need to decompose the rational function into partial fractions. The denominator, x^3 - 1, can be factored using the difference of cubes formula.
x^3 - 1 = (x - 1)(x^2 + x + 1)
To decompose the rational function, we write it as a sum of two or more fractions with simpler denominators. Let's assume the decomposition to be:
(8x - 5)/(x^3 - 1) = A/(x - 1) + B/(x^2 + x + 1)
To find the values of A and B, we need to clear the denominators by multiplying through by the common denominator, which is (x - 1)(x^2 + x + 1):
8x - 5 = A(x^2 + x + 1) + B(x - 1)
Now, we can expand and simplify the equation:
8x - 5 = Ax^2 + Ax + A + Bx - B
Comparing the coefficients of like terms on both sides of the equation, we get the following system of equations:
1. Coefficient of x^2: A = 0
2. Coefficient of x: A + B = 8
3. Coefficient of constant term: A - B = -5
From the first equation, we find that A = 0. Substituting this into the second and third equations, we get:
0 + B = 8
0 - B = -5
Solving these equations, we find B = 8 and A = 0.
Now that we have determined the values of A and B, we can rewrite the integral as:
∫ ((8x - 5)/(x^3 - 1)) dx = ∫ (A/(x - 1) + B/(x^2 + x + 1)) dx
= ∫ (8/(x^2 + x + 1)) dx
The integral of 8/(x^2 + x + 1) can be solved using various methods such as completing the square, substitution, or recognizing it as a standard integral.
Using the method of completing the square, we complete the square in the denominator:
x^2 + x + 1 = (x + 1/2)^2 + 3/4
Now, we can rewrite the integral as:
∫ (8/(x^2 + x + 1)) dx = ∫ (8/((x + 1/2)^2 + 3/4)) dx
At this point, you can let u = x + 1/2 to simplify the integral further. The integral becomes:
∫ (8/(u^2 + 3/4)) du
From here, you can use a simple substitution to solve the integral.
I hope this explanation helps you decompose and solve the given integral!