Use part I of the Fundamental Theorem of Calculus to find the derivative of
g(x)= integrate from 9x to 4x of ((u+4)/(u-1))du
since (u+4)/(u-1) = 1 + 5/(u-1)
integral = u + 5log u [9x,4x]
-5(x - log(4x-1) + log(9x-1))
To find the derivative of the function g(x) using part I of the Fundamental Theorem of Calculus, we can follow these steps:
Step 1: Rewrite the integral notation using appropriate limits:
g(x) = ∫[9x to 4x] ((u+4)/(u-1)) du
Step 2: Determine the antiderivative of the integrand:
Integrate the function ((u+4)/(u-1)) with respect to u, treating x as a constant. Let's call this antiderivative F(u):
F(u) = ∫ ((u+4)/(u-1)) du
Step 3: Apply the Fundamental Theorem of Calculus - Part I:
According to the theorem, if F(u) is an antiderivative of ((u+4)/(u-1)), then the derivative of the integral from a constant 'a' to the variable 'x' of ((u+4)/(u-1)) du is equal to F(x) - F(a), where F(x) is the antiderivative at x and F(a) is the antiderivative at a.
Step 4: Evaluate F(x) and F(9x):
To evaluate F(x), plug x into the antiderivative F(u) and obtain a new function F(x). To evaluate F(9x), substitute 9x for u in the antiderivative F(u) and simplify.
Step 5: Calculate the derivative by using F(x) - F(a):
Finally, calculate the derivative of g(x) by subtracting F(9x) and F(4x), since those are the limits of integration mentioned in the original function g(x).
Following these steps, you should be able to find the derivative of the given function using the Fundamental Theorem of Calculus - Part I.