Yesterday I posted this number that I don't understand. There was a mistake in the question,this is the right one:
∫ (5^(3/x) - (x^8 + 9)^1/3 ) / 3x^2
will u, for sub. be = x^8 + 9?
Then du is = 8x^7 dx
x^7 dx = 1/8 du
Thank you!
As Steve mentioned yesterday for this same question, the 5^(3/x) is a stopper.
I put your expression in the Wolfram integrator with the necessary changes in brackets, and frankly I don't even want to understand their answer.
http://integrals.wolfram.com/index.jsp?expr=%285%5E%283%2Fx%29+-+%28x%5E8+%2B+9%29%5E%281%2F3%29+%29+%2F+%283x%5E2%29&random=false
Actually, I have to reconsider.
If u = 5^(3/x)
du = ln5 5^(3/x) (-3/x^2)
we have the form, and we end up with
∫ (5^(3/x) - (x^8 + 9)^1/3 ) / 3x^2 dx
= -1/(9ln5) ∫ u - (x^8 + 9)^1/3 ) du
The show stopper is that ∛(x^8+9)
Thank you both for taking your time to try this :)
To solve the given integral ∫ (5^(3/x) - (x^8 + 9)^(1/3)) / (3x^2), we can make a substitution. Let's set u = x^8 + 9. Now, we need to find du in terms of x.
To find du, we take the derivative of u with respect to x: du/dx = 8x^7.
Now, we can rewrite the integral with the substitution:
∫ (5^(3/x) - (x^8 + 9)^(1/3)) / (3x^2) dx = ∫ (5^(3/x) - u^(1/3)) / (3x^2) dx
To simplify this further, let's rewrite dx in terms of du:
dx = (1/8x^7) du
Now, substitute this into the integral:
∫ (5^(3/x) - u^(1/3)) / (3x^2) (1/8x^7) du
Simplify further:
(1/(24x^9)) ∫ (5^(3/x) - u^(1/3)) du
Now, we can integrate with respect to u:
(1/(24x^9)) [(3/x) ln(5) - (3/4) u^(4/3)] + C
Finally, substitute back u = x^8 + 9:
(1/(24x^9)) [(3/x) ln(5) - (3/4) (x^8 + 9)^(4/3)] + C
And that is the solution to the integral.