What is the integral of dx/(x^3-9x)? Please post steps too please!
To find the integral of dx/(x^3-9x), we can start by factoring the denominator.
x^3 - 9x = x(x^2 - 9)
Next, we can further simplify the expression:
x(x^2 - 9) = x(x - 3)(x + 3)
Now, we can rewrite the integral as follows:
∫ dx/(x^3-9x) = ∫ dx/(x(x - 3)(x + 3))
The integral is now in the form of a partial fraction. To solve this, we need to find the values of A, B, and C in the following expression:
dx/(x(x - 3)(x + 3)) = A/x + B/(x - 3) + C/(x + 3)
To find the values of A, B, and C, we need to clear the denominator by multiplying both sides of the equation by (x)(x - 3)(x + 3):
1 = A(x - 3)(x + 3) + B(x)(x + 3) + C(x)(x - 3)
Now, we can determine the values of A, B, and C by substituting specific values for x:
For x = 0:
1 = -9A
A = -1/9
For x = 3:
1 = 6B
B = 1/6
For x = -3:
1 = 6C
C = 1/6
Now that we have the values of A, B, and C, we can rewrite the integral as a sum of three simpler integrals:
∫ dx/(x(x - 3)(x + 3)) = ∫ -1/(9x) dx + ∫ 1/(6(x - 3)) dx + ∫1/(6(x + 3)) dx
Finally, we can integrate each term individually:
∫ dx/(x(x - 3)(x + 3)) = (-1/9) ln|x| + (1/6) ln|x - 3| + (1/6) ln|x + 3| + C
Therefore, the integral of dx/(x^3-9x) is given by:
(-1/9) ln|x| + (1/6) ln|x - 3| + (1/6) ln|x + 3| + C