(1 point) Write an expression for the function, f(x), with the properties. f′(x)=(ln(x)2)/x and f(3)=7.
f(x)=∫t=b
t=a
Oh, the function is in integral form. How fancy! Allow me to simplify it for you.
Given that f'(x) = (ln(x)^2) / x, we need to find f(x) using this information.
To do this, we can use the fundamental theorem of calculus:
f(x) = ∫ (ln(t)^2) / t dt
Now we just need to determine the limits of integration, a and b. Thankfully, we are given that f(3) = 7. So, we can set:
7 = ∫ (ln(t)^2) / t dt, where t ranges from a to 3.
And that's it! We have the expression for f(x) in integral form.
To find an expression for the function f(x) with the given properties, we can integrate f'(x) and use the information about f(3).
Given: f′(x) = (ln(x)^2) / x and f(3) = 7.
To find f(x), we need to integrate f'(x):
∫f′(x) dx = ∫(ln(x)^2) / x dx
Using the power rule for integration, we can rewrite the integral as:
∫(ln(x)^2) / x dx = ∫ln(x)^2 * x^(-1) dx
Now, using integration by parts, we can write:
∫u dv = uv - ∫v du
Let's define u as ln(x) and dv as x^(-1) dx. Then, we can find du and v.
du = (1/x) dx
v = ∫x^(-1) dx = ln|x|
Using these values, we can rewrite the integral as:
∫ln(x)^2 * x^(-1) dx = ln(x) * ln|x| - ∫ln|x| * (1/x) dx
Now, we can integrate the remaining part of the integral:
∫ln|x| * (1/x) dx = ∫(ln|x| / x) dx
Again, using integration by parts, we define u as ln|x| and dv as x^(-1) dx. Then, we can find du and v.
du = (1/x) dx
v = ∫x^(-1) dx = ln|x|
Using these values, we can rewrite the integral as:
∫(ln|x| / x) dx = ln|x| * ln|x| - ∫ln|x| * (1/x) dx
Now, substituting back the result of the previous integral, we have:
∫ln(x)^2 * x^(-1) dx = ln(x) * ln|x| - ln|x| * ln|x| + ∫ln|x| * (1/x) dx
Simplifying, we get:
∫ln(x)^2 * x^(-1) dx = ln(x) * ln|x| - ln^2 |x| + ∫ln|x| * (1/x) dx
Now, we can substitute the limits of integration, which are not given in the question. Let's assume the limits are from a to b:
∫t=b
t=a ln(x)^2 * x^(-1) dx = [ln(x) * ln|x| - ln^2 |x| + ∫ln|x| * (1/x) dx] evaluated from b to a
Finally, we substitute the given value f(3) = 7, so x=3:
f(3) = ∫t=3
t=a ln(x)^2 * x^(-1) dx = [ln(x) * ln|x| - ln^2 |x| + ∫ln|x| * (1/x) dx] evaluated from 3 to a = 7
Unfortunately, without the specific limits of integration (a, b) or further information, we cannot find an explicit expression for f(x) in terms of x.
To find an expression for the function f(x) using the given properties, we need to integrate the derivative f'(x) and use the initial condition f(3) = 7.
First, let's integrate f'(x) to find f(x). The integral of (ln(x)^2)/x with respect to x is given by:
∫(ln(x)^2)/x dx
We can rewrite the expression as:
∫ln(x)^2/x dx
To integrate this expression, we use substitution. Let u = ln(x), then du/dx = 1/x, and dx = x du. Substituting these values, we get:
∫u^2 du
Integrating further, we have:
(1/3) u^3 + C
Substituting back u = ln(x), we get:
(1/3) (ln(x))^3 + C
Now, to find the value of C, we use the initial condition f(3) = 7. Substituting x = 3 into the expression for f(x), we have:
(1/3) (ln(3))^3 + C = 7
Solving for C, we subtract (1/3) (ln(3))^3 from both sides:
C = 7 - (1/3) (ln(3))^3
Therefore, the expression for the function f(x) is:
f(x) = (1/3) (ln(x))^3 + 7 - (1/3) (ln(3))^3
f′(x)=ln^2(x)/x
let u=lnx and you have f'(x) = u^2 du
f(x) = 1/3 ln^3(x)+C
f(3)=7, so
1/3 ln^3(3)+C = 7
C = 7 - 1/3 ln^3(3)
f(x) = 1/3 ln^3(x) + 7 - 1/3 ln^3(3) ≈ 1/3 ln^3(x) + 6.558