is the integral of 2x/(x^3-x)
=-1/2ln(x+1)+1/2ln(x-1)
I got this by doing this:
(\=integral sign)
2\1/(x^2-1)
2\1/(x+1)(x-1)
=A/(x+1)+B/(x-1)
=Ax-A+Bx+B=1
Ax+Bx=0
B-A=1
B=1+A
(1+A)x+Ax=0
x+Ax+Ax=0
1+AB=0
2A=-1
A=-1/2
B+1/2=1, B=1/2
etc.
I differ by a factor of 2
2[ A/(x+1) +B/(x-1) ]
A = -1/2, B = +1/2
so
2 int [ -.5 dx/(x+1) + .5 dx /(x-1) ]
1 int [ dx/(x+1) + dx/(x-1) ]
ln (x+1) + ln (x-1)
which is
ln[ (x+1)/(x-1) ]
To see if your answer is correct, take the derivative. Itis
-(1/2)/(x+1) +(1/2)/(x-1)
= [-(1/2)(x-1) + (1/2)(x+1)]/(x^2 -1)
= 1/(x^2-1)
Your riginal integrand is equal to 2/(x^2-1), so your answer seems to be off by a factor of 2.
Sorry
ln (x+1)(x-1)
which is
ln(x^2-1)
Which I should have seen in the first place.
took ya long enuff
Yeah, but I am really old :)
To find the integral of the function 2x/(x³ - x), you first need to decompose the function into partial fractions. Here's how you do it:
1. Start with the original function: 2x/(x³ - x).
2. Factor the denominator: x³ - x = x(x² - 1) = x(x + 1)(x - 1).
3. Write the partial fraction decomposition as A/(x + 1) + B/(x - 1).
4. Multiply both sides of the equation by the common denominator (x + 1)(x - 1) to get rid of the fractions.
You will have 2x = A(x - 1) + B(x + 1).
5. Expand the equation: 2x = Ax - A + Bx + B.
6. Combine like terms: 2x = (A + B)x + (-A + B).
7. Equate the coefficients of x: 2 = A + B.
8. Equate the constant terms: 0 = -A + B.
9. Solve the system of equations to find A and B.
By adding the two equations, we get 2 + 0 = (A + B) + (-A + B).
Simplifying further, we get 2 = 2B, which implies B = 1.
Substituting B = 1 into any of the previous equations, we get -A + 1 = 0, which implies A = -1.
10. Now that we have A = -1 and B = 1, we can rewrite the function as: -1/(x + 1) + 1/(x - 1).
Now, let's integrate the function:
1. The integral of -1/(x + 1) can be written as -ln| x + 1 |.
2. Similarly, the integral of 1/(x - 1) can be written as ln| x - 1 |.
3. Finally, the integral of 2x/(x³ - x) is: -ln| x + 1 | + ln| x - 1 |, which can be simplified using logarithmic properties to give:
ln| (x - 1)/(x + 1) |.
Therefore, the given integral of 2x/(x³ - x) is not equal to -1/2ln(x + 1) + 1/2ln(x - 1). It is equal to ln| (x - 1)/(x + 1) |.