Question

Let $f : (0, \infty) \to \mathbb{R}$ and $F(x) = \int_{0}^{x} tf(t) \, dt$. If $F(x^2) = x^4 + x^5$, then $\sum_{r=1}^{12} f(r^2)$is equal to:

Answer 1

Step 1. Fundamental Theorem of Calculus and the Given Information:

We are given that $F(x) = \int_0^x t \cdot f(t) dt$. The Fundamental Theorem of Calculus states that the derivative of a definite integral with respect to its upper limit is the integrand evaluated at that upper limit. Therefore, we have:

$F'(x) = x \cdot f(x)$

We're also given that $F(x^2) = x^4 + x^5$. Let's substitute $t = x^2$:

$F(t) = t^2 + t^{5/2}$

Step 2. Finding f(t):

Now we differentiate $F(t)$ to find $F'(t)$:

$F'(t) = \frac{d}{dt}(t^2 + t^{5/2}) = 2t + \frac{5}{2}t^{3/2}$

Since $F'(t) = t \cdot f(t)$, we can solve for $f(t)$:

$t \cdot f(t) = 2t + \frac{5}{2}t^{3/2}$

$f(t) = \frac{2t + \frac{5}{2}t^{3/2}}{t} = 2 + \frac{5}{2}t^{1/2}$

Step 3. Evaluating the Summation:

We want to find $\sum_{r=1}^{12} f(r^2)$. Substituting our expression for $f(t)$:

$\sum_{r=1}^{12} f(r^2) = \sum_{r=1}^{12} \left( 2 + \frac{5}{2}r \right)$

Step 4. Summation Properties and Simplification:

We can split the summation into two separate sums:

$\sum_{r=1}^{12} f(r^2) = \sum_{r=1}^{12} 2 + \frac{5}{2} \sum_{r=1}^{12} r$

The first term is simply $2$ added 12 times: $2 \times 12 = 24$. The second term is the sum of the integers from 1 to 12, which can be calculated using the formula for the sum of an arithmetic series: $\sum_{r=1}^n r = \frac{n(n+1)}{2}$.

$\sum_{r=1}^{12} r = \frac{12(12+1)}{2} = \frac{12(13)}{2} = 78$

Step 5. Final Calculation:

Substituting back into our equation:

$\sum_{r=1}^{12} f(r^2) = 24 + \frac{5}{2} (78) = 24 + 5(39) = 24 + 195 = 219$

Therefore, the final answer is:

$\sum_{r=1}^{12} f(r^2) = 219$

Final Answer: The final answer is $\boxed{219}$

Answer 2

Step 1. Fundamental Theorem of Calculus and the Given Information:

We are given that $F(x) = \int_0^x t \cdot f(t) dt$. The Fundamental Theorem of Calculus states that the derivative of a definite integral with respect to its upper limit is the integrand evaluated at that upper limit. Therefore, we have:

$F'(x) = x \cdot f(x)$

We're also given that $F(x^2) = x^4 + x^5$. Let's substitute $t = x^2$:

$F(t) = t^2 + t^{5/2}$

Step 2. Finding f(t):

Now we differentiate $F(t)$ to find $F'(t)$:

$F'(t) = \frac{d}{dt}(t^2 + t^{5/2}) = 2t + \frac{5}{2}t^{3/2}$

Since $F'(t) = t \cdot f(t)$, we can solve for $f(t)$:

$t \cdot f(t) = 2t + \frac{5}{2}t^{3/2}$

$f(t) = \frac{2t + \frac{5}{2}t^{3/2}}{t} = 2 + \frac{5}{2}t^{1/2}$

Step 3. Evaluating the Summation:

We want to find $\sum_{r=1}^{12} f(r^2)$. Substituting our expression for $f(t)$:

$\sum_{r=1}^{12} f(r^2) = \sum_{r=1}^{12} \left( 2 + \frac{5}{2}r \right)$

Step 4. Summation Properties and Simplification:

We can split the summation into two separate sums:

$\sum_{r=1}^{12} f(r^2) = \sum_{r=1}^{12} 2 + \frac{5}{2} \sum_{r=1}^{12} r$

The first term is simply $2$ added 12 times: $2 \times 12 = 24$. The second term is the sum of the integers from 1 to 12, which can be calculated using the formula for the sum of an arithmetic series: $\sum_{r=1}^n r = \frac{n(n+1)}{2}$.

$\sum_{r=1}^{12} r = \frac{12(12+1)}{2} = \frac{12(13)}{2} = 78$

Step 5. Final Calculation:

Substituting back into our equation:

$\sum_{r=1}^{12} f(r^2) = 24 + \frac{5}{2} (78) = 24 + 5(39) = 24 + 195 = 219$

Therefore, the final answer is:

$\sum_{r=1}^{12} f(r^2) = 219$

Final Answer: The final answer is $\boxed{219}$