If Upper F(x) = int_1^x (f(t)) d t where f(t) = int_1^(t^2) (sqrt(2 + u^2) divided by u) d u find (Upeer F) prime prime (2).

Question

​

a.
3 sqrt(3)

b.
3 sqrt(2)

c.
(3 sqrt(2)) over 2

d.
6 sqrt(2)

e.
6 sqrt(3)

Answer 1

To find (Upper F) prime prime (2), we first need to find Upper F(x).

Given that Upper F(x) = int_1^x (f(t)) dt where f(t) = int_1^(t^2) (sqrt(2 + u^2) / u) du, we first find f(t):

f(t) = int_1^(t^2) (sqrt(2 + u^2) / u) du

Let's simplify f(t) by substituting u = t:

f(t) = int_1^(t^2) (sqrt(2 + t^2) / t) du

Now, we can rewrite Upper F(x) in terms of f(t):

Upper F(x) = int_1^x f(t) dt

Upper F(x) = int_1^x [int_1^(t^2) (sqrt(2 + u^2) / u) du] dt

Now, let's find (Upper F) prime prime (2):

(Upper F) prime prime (2) = f(2)

f(2) = int_1^(2^2) (sqrt(2 + u^2) / u) du

f(2) = int_1^4 (sqrt(2 + u^2) / u) du

This integral needs to be solved to find the value of f(2), which will result in the answer to (Upper F) prime prime (2).

Therefore, without solving the integral, the answer cannot be determined precisely. The solution involves integrating the given expression from 1 to 4.