Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)
f ''(x) = 5/6x^5/6
if f'(x) = ax^n
then f(x) = a (1/(n+1)) x^(n+1) + c
apply
To find the antiderivative of f ''(x) = 5/6x^(5/6), we need to integrate it twice.
First, we integrate f ''(x) with respect to x to find f '(x):
∫ (f ''(x)) dx = ∫ (5/6x^(5/6)) dx
To integrate x^(5/6), we can use the power rule for integration. It states that if we have a function of the form x^n, where n is any real number except -1, the antiderivative is (1/(n+1)) * x^(n+1).
Using the power rule, we integrate x^(5/6) as follows:
∫ (5/6x^(5/6)) dx = (5/6) * (x^(5/6 + 1))/(5/6 + 1) + C1 = (5/6) * (x^(11/6))/(11/6) + C1 = (5/6) * (6/11) * x^(11/6) + C1 = (5/11) * x^(11/6) + C1
Where C1 is the constant of integration.
Now, we have f '(x) = (5/11) * x^(11/6) + C1.
Next, we need to integrate f '(x) to find f(x):
∫ (f '(x)) dx = ∫ [(5/11) * x^(11/6) + C1] dx
Using the power rule again, we integrate x^(11/6):
∫ (5/11) * x^(11/6) dx = (5/11) * (x^(11/6 + 1))/(11/6 + 1) + C2 = (5/11) * (x^(17/6))/(17/6) + C2 = (5/11) * (6/17) * x^(17/6) + C2 = (5/17) * x^(17/6) + C2
Where C2 is the constant of integration.
Finally, we have f(x) = (5/17) * x^(17/6) + C2.
Therefore, the function f(x) is (5/17) * x^(17/6) + C2, where C2 is the constant of integration.