Final answer:
To solve ∫x^2-36√x dx, separate the integral into ∫x^2 dx and ∫-36x^(1/2) dx, integrate each term to get (1/3)x^3 for the first and 24x^(3/2) for the second, and combine to find the antiderivative (1/3)x^3 - 24x^(3/2) + C.
Step-by-step explanation:
To find the integral of ∫x^2 - 36√x dx for x>6, we first need to simplify the given expression. Recognizing that √x is the same as x^(1/2), we can rewrite the integral as:
∫(x^2 - 36x^(1/2)) dx
With this rewriting, we can separate the integral into two parts:
∫x^2 dx - ∫36x^(1/2) dx
Now we can integrate each term separately:
The integral of x^2 with respect to x is (1/3)x^3, and the integral of 36x^(1/2) with respect to x is 36*(2/3)x^(3/2) which simplifies to 24x^(3/2).
Combining these results, the antiderivative is:
(1/3)x^3 - 24x^(3/2) + C
Where C is the constant of integration.
Therefore, the solution to the integral is (1/3)x^3 - 24x^(3/2) + C.