Antidifferentiate: ∫ (-2/3)⁢*(1/ln⁡(x^3⁢)*x) ⁢ⅆx (Calculus)?

Question

Can you help me with this problem? I don't want the answer, just get me started. Thanks!

Answer 1

∫ (-2/3)⁢*(1/ln⁡(x^3⁢)*x) ⁢ⅆx

First, pull out all constants; constants get in the way of things.

(-2/3) * ∫ (1/[ln⁡(x^3⁢)*x] dx

Next, note that ln(x^3) is the same as 3ln(x). This is by one of the properties of logarithmic functions. So now we have:

(-2/3) * ∫ (1/[(3ln⁡(x))*x] dx

Again, pull out messy constants (in this case, the 3).

(-2/9) * ∫ (1/[(ln⁡(x))*x] dx

Let's change it around a bit:

(-2/9) * ∫ (1/(ln⁡(x)) (1/x) dx

Now, we use substitution.
let u = ln(x)
du = 1/x dx

The reason why I split it up above is to make it obvious that du replaced (1/x dx), as we just solved. So, we do the appropriate replacement.

(-2/9) * ∫ (1/u) du

And now the integral is trivial.

(-2/9) * ln|u| + C

Replace u = ln(x) back, to get
(-2/9) * ln | ln(x) | + C

Answer 2

ok Lis---improve the r(x) to get x^5 -4x^3. Now take derivatives of each term. 5x^4 and -12x^2. Now recombine to get r'(x) = x^2*(5x^2-12). you additionally can use the product rule however the above is extra uncomplicated.