f'(x)=In(25x)/x between x =1 amd x=12 12. f(x)= ??
To find the value of f(x) using the given derivative f'(x) = ln(25x)/x, we need to integrate the derivative to obtain the original function f(x). Here's how to do it step-by-step:
Step 1: Recognize that the given expression is the derivative of f(x).
Step 2: Integrate f'(x) with respect to x. To integrate ln(25x)/x, use the method of integration by parts.
Let's set u = ln(25x) and dv = 1/x. Then, du = (1/x) dx and v = ln|x|. Applying the integration by parts formula, we have:
∫ [(ln(25x)/x) dx] = ∫ u dv
= uv - ∫ v du
= ln|x| * ln(25x) - ∫ ln|x| * (1/x) dx
Step 3: Integrate the remaining integral on the right using substitution.
Now, let's simplify the remaining integral: ∫ ln|x| * (1/x) dx.
To resolve this, let u = ln|x| and du = (1/x) dx. Rearranging, we have x = e^u.
Substituting these values into the integral, we get:
∫ ln|x| * (1/x) dx = ∫ u du = (1/2)u^2 + C, where C is the constant of integration.
Step 4: Substitute the value of u back in terms of x.
Since u = ln|x|, we substitute ln|x| back into the expression:
(1/2)u^2 + C = (1/2)(ln|x|)^2 + C
Step 5: Evaluate the definite integral between x = 1 and x = 12.
To find f(x), we need to evaluate the definite integral of f'(x) between the limits x = 1 and x = 12.
Using the Antiderivative Fundamental Theorem of Calculus, we evaluate the integral at x = 12 and subtract the value of the integral at x = 1:
f(x) = ∫[1 to 12] ln(25x)/x dx
= [(1/2)(ln|x|)^2 + C] from 1 to 12
= [(1/2)(ln|12|)^2 + C] - [(1/2)(ln|1|)^2 + C]
Since C is a constant, it cancels out:
f(x) = (1/2)(ln|12|)^2 - (1/2)(ln|1|)^2
= (1/2)(ln(12))^2 - (1/2)(ln(1))^2
= (1/2)(ln(12))^2 - (1/2)(ln(1))^2
= (1/2)(ln(12))^2 - 0
= (1/2)(ln(12))^2
Therefore, f(x) = (1/2)(ln(12))^2.