integrate:dx/(3x^3-5)^3

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To integrate the expression ∫(dx/(3x^3-5)^3), you can use a standard integration technique. One approach is to use a trigonometric substitution. Here's how you can do it step by step:

Step 1: Start by performing the substitution:
Let 3x^3-5 = tan(theta)
Differentiating both sides with respect to x:
9x^2 dx = sec^2(theta) d(theta)
Therefore, dx = sec^2(theta)/(9x^2) d(theta)

Step 2: Rewrite the integral in terms of theta:
∫(dx/(3x^3-5)^3) = ∫(sec^2(theta)/(9x^2*(tan(theta))^3) d(theta)
= 1/9 * ∫(sec^2(theta)/(tan(theta))^3) d(theta)

Step 3: Simplify the integrand using trigonometric identities:
Using the identity sec^2(theta) = 1 + tan^2(theta), and replacing (tan(theta))^3 with (3x^3-5)^3, the integrand becomes:
1/9 * (1/(1 + (3x^3 - 5)^2)) d(theta)

Step 4: Convert the integral back to x:
Using the original substitution, x = (tan(theta) + 5/3)^(1/3), you can express d(theta) in terms of dx:
d(theta) = dx/(sec^2(theta)/(9x^2))
= 9x^2 dx/(1 + (3x^3 - 5)^2)

Step 5: Apply the substitution to the integral:
∫(dx/(3x^3-5)^3) = 1/9 ∫((9x^2 dx)/(1 + (3x^3 - 5)^2))
= ∫((9x^2 dx)/(9x^6 - 30x^3 + 26))

Step 6: Solve the integral:
The integral of 9x^2 / (9x^6 - 30x^3 + 26) can be solved using partial fractions, polynomial division, or computer algebra systems like Wolfram Alpha.

However, the integration process for this expression can be quite complex and involve symbolic manipulation. It's recommended to use specialized software or tools to compute this integral.