Hello, this is the last question from my homework that I get stuck at:
∫ (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
from there I'm stuck.
Thank you
I suspect you have a typo. Go to
wolframalpha . com and type in
integral (5^3/x - (x^8 + 9)^1/3 ) / 3x^2 dx
Look at how it interprets it, and add parentheses till you get what you want.
That 5^3/x is a killer, especially if you meant 5^(3/x)!
I did mean 5^(3/x), sorry about that!
To integrate the given expression ∫ (5^3/x - (x^8 + 9)^1/3 ) / 3x^2, you are on the right track with your substitution. Let's go through the steps together.
Step 1: Perform the substitution
Let u = x^8 + 9. This means du = 8x^7 dx.
Rearranging for dx, you have dx = du / (8x^7).
Step 2: Adjust the integrand using the substitution
Using the substitution, the integrand becomes:
(5^3/x - (u)^(1/3)) / (3x^2).
Now, replace x^2 with (1/8)(du/x^7) according to the rearranged expression for dx.
Step 3: Simplify the expression
The integrand becomes:
(125/x - u^(1/3)) / (3x^2 * 8x^7).
Simplify further by combining the terms in the denominator:
(125 - 8x^5 * (u)^(1/3)) / (24x^9).
Step 4: Distribute the factor of 8x^5 inside the cube root
Using the properties of exponents, the expression becomes:
(125 - 8(u^(1/3))(x^5)) / (24x^9).
Step 5: Substitute back for u
Recall that u = x^8 + 9, so the expression becomes:
(125 - 8((x^8 + 9)^(1/3))(x^5)) / (24x^9).
Step 6: Evaluate the integral
Now, you can integrate the simplified expression:
∫ (125 - 8((x^8 + 9)^(1/3))(x^5)) / (24x^9) dx.
Solving this integral requires integration techniques such as u-substitution or integration by parts. However, the specific method may depend on the context or further instructions provided in your homework.