`f:(0,infty)rarrR and F(x)=int_(0)^(x)t f(t)dt`
If `F(x^(2))=x^(4)+x^(5),"then" Sigma_(r=1)^(12)f(r^(2))` is equal to S

Options

A. 216
B. 219
C. 221
D. 223

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the given function
We are given a function \( F(x) \) defined as:
\[ F(x) = \int_0^x t f(t) dt \]
and we know that:
\[ F(x^2) = x^4 + x^5 \]

### Step 2: Differentiate both sides
To find \( f(x) \), we will differentiate both sides of the equation \( F(x^2) = x^4 + x^5 \) with respect to \( x \).

Using the chain rule and Leibniz rule for differentiation under the integral sign:
\[ \frac{d}{dx} F(x^2) = \frac{d}{dx} \left( \int_0^{x^2} t f(t) dt \right) = x^2 f(x^2) \cdot 2x = 2x^3 f(x^2) \]

Differentiating the right-hand side:
\[ \frac{d}{dx}(x^4 + x^5) = 4x^3 + 5x^4 \]

### Step 3: Set the derivatives equal
Now we equate the two results:
\[ 2x^3 f(x^2) = 4x^3 + 5x^4 \]

### Step 4: Solve for \( f(x^2) \)
Dividing both sides by \( 2x^3 \) (assuming \( x \neq 0 \)):
\[ f(x^2) = \frac{4x^3 + 5x^4}{2x^3} = 2 + \frac{5}{2} x \]

### Step 5: Find \( f(x) \)
Since \( f(x^2) = 2 + \frac{5}{2} x \), we can express \( f(x) \) by substituting \( x \) with \( \sqrt{x} \):
\[ f(x) = 2 + \frac{5}{2} \sqrt{x} \]

### Step 6: Calculate \( \Sigma_{r=1}^{12} f(r^2) \)
Now we need to compute:
\[ \Sigma_{r=1}^{12} f(r^2) = \Sigma_{r=1}^{12} \left( 2 + \frac{5}{2} r \right) \]
This can be separated into two summations:
\[ = \Sigma_{r=1}^{12} 2 + \Sigma_{r=1}^{12} \frac{5}{2} r \]

Calculating the first summation:
\[ \Sigma_{r=1}^{12} 2 = 2 \times 12 = 24 \]

Calculating the second summation using the formula for the sum of the first \( n \) natural numbers:
\[ \Sigma_{r=1}^{12} r = \frac{12 \times 13}{2} = 78 \]
Thus:
\[ \Sigma_{r=1}^{12} \frac{5}{2} r = \frac{5}{2} \times 78 = 195 \]

### Step 7: Combine the results
Now we combine both results:
\[ \Sigma_{r=1}^{12} f(r^2) = 24 + 195 = 219 \]

### Final Answer
Thus, the value of \( S \) is:
\[ \boxed{219} \]