Question

Let \( f : [1, \infty) \to [2, \infty) \) be a differentiable function. If

\( 10 \int_{1}^{x} f(t) \, dt = 5x f(x) - x^5 - 9 \) for all \( x \ge 1 \), then the value of \( f(3) \) is ______.

• A. 18
• B. 32
• C. 22
• D. 20

Hint:

For this type of problem, use the properties of definite integrals and apply the fundamental theorem of calculus along with the chain rule. Simplify the equation step by step for easy solution.

Answer 1

The Correct Option is B

We are given a differentiable function \( f(x) \) defined by an integral equation, and we need to find the value of this function at \( x = 3 \).

Concept Used:

To solve this problem, we will use the following key concepts:

1. Leibniz Rule (Fundamental Theorem of Calculus, Part 1): This rule is used to differentiate an integral with a variable upper limit. \[ \frac{d}{dx} \int_{a}^{x} g(t) \, dt = g(x) \]
2. Solving First-Order Linear Differential Equations: An equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \) can be solved using an integrating factor (I.F.).
   • The Integrating Factor is given by \( \text{I.F.} = e^{\int P(x) \, dx} \).
   • The solution is given by \( y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) \, dx + C \), where C is the constant of integration.

The strategy is to first convert the given integral equation into a differential equation by differentiating it, then solve the differential equation to find the function \( f(x) \), and finally calculate \( f(3) \).

Step-by-Step Solution:

Step 1: Differentiate the given integral equation with respect to \( x \).

The given equation is:

\[ 10 \int_{1}^{x} f(t) \, dt = 5x f(x) - x^5 - 9 \]

Differentiating both sides with respect to \( x \) using the Leibniz rule for the left side and the product rule for the term \( 5x f(x) \) on the right side:

\[ \frac{d}{dx} \left( 10 \int_{1}^{x} f(t) \, dt \right) = \frac{d}{dx} \left( 5x f(x) - x^5 - 9 \right) \] \[ 10 f(x) = \left( 5 \cdot f(x) + 5x \cdot f'(x) \right) - 5x^4 - 0 \]

Step 2: Rearrange the resulting equation to form a first-order linear differential equation.

\[ 10 f(x) = 5f(x) + 5x f'(x) - 5x^4 \]

Simplifying the equation:

\[ 5 f(x) = 5x f'(x) - 5x^4 \]

Dividing the entire equation by 5:

\[ f(x) = x f'(x) - x^4 \]

Rearranging it into the standard form \( f'(x) + P(x)f(x) = Q(x) \):

\[ x f'(x) - f(x) = x^4 \]

Since \( x \geq 1 \), we can divide by \( x \):

\[ f'(x) - \frac{1}{x} f(x) = x^3 \]

Step 3: Calculate the integrating factor (I.F.) for this linear differential equation.

Here, \( P(x) = -\frac{1}{x} \). The integrating factor is:

\[ \text{I.F.} = e^{\int P(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} \]

Since \( x \geq 1 \), \( |x| = x \). So,

\[ \text{I.F.} = e^{-\ln x} = e^{\ln(x^{-1})} = x^{-1} = \frac{1}{x} \]

Step 4: Find the general solution of the differential equation.

The solution is given by \( f(x) \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) \, dx + C \). With \( Q(x) = x^3 \):

\[ f(x) \cdot \frac{1}{x} = \int x^3 \cdot \frac{1}{x} \, dx + C \] \[ \frac{f(x)}{x} = \int x^2 \, dx + C \] \[ \frac{f(x)}{x} = \frac{x^3}{3} + C \]

The general solution for \( f(x) \) is:

\[ f(x) = \frac{x^4}{3} + Cx \]

Step 5: Determine the value of the integration constant C.

We use the original integral equation and substitute \( x=1 \) to find an initial condition.

\[ 10 \int_{1}^{1} f(t) \, dt = 5(1) f(1) - (1)^5 - 9 \]

Since the integral from 1 to 1 is zero:

\[ 0 = 5f(1) - 1 - 9 \] \[ 0 = 5f(1) - 10 \implies 5f(1) = 10 \implies f(1) = 2 \]

Now, substitute \( x=1 \) and \( f(1)=2 \) into our general solution:

\[ 2 = \frac{(1)^4}{3} + C(1) \] \[ 2 = \frac{1}{3} + C \] \[ C = 2 - \frac{1}{3} = \frac{5}{3} \]

Step 6: Write the particular solution for \( f(x) \).

Substituting \( C = 5/3 \) back into the general solution, we get the specific function:

\[ f(x) = \frac{x^4}{3} + \frac{5}{3}x \]

Final Computation & Result

We are asked to find the value of \( f(3) \). We substitute \( x=3 \) into the function we found:

\[ f(3) = \frac{(3)^4}{3} + \frac{5}{3}(3) \] \[ f(3) = \frac{81}{3} + 5 \] \[ f(3) = 27 + 5 = 32 \]

Thus, the value of \( f(3) \) is 32.

Answer 2

\[ 10 \frac{d}{dx} \int_1^x f(t) \, dt = \frac{d}{dx} \left( 5x f(x) - x^5 - 9 \right) \] \[ \Rightarrow 10 f(x) = 5f(x) + 5x f'(x) - 5x^4 \] \[ \Rightarrow f(x) + x^4 = x f'(x) \] \[ \Rightarrow \frac{dy}{dx} + y \left( \frac{1}{x} \right) = x^3 \] \[ \Rightarrow \frac{dy}{dx} = x^3 - y \left( \frac{1}{x} \right) \] \[ \Rightarrow y e^{-\frac{1}{x}} = \int x^3 e^{\frac{1}{x}} \, dx + c \] \[ \Rightarrow y = \frac{y}{x} \cdot \int x^3 \, dx \] \[ \Rightarrow y = \frac{x^3}{3} + c \] \[ \Rightarrow y = \frac{x^3}{3} + c \] \[ \text{Put } x = 1 \text{ in the given equation:} \] \[ 0 = 5f(1) - 9 - 9 \] \[ f(1) = 2 \Rightarrow c = \frac{5}{3} \] \[ f(3) = \frac{27}{3} + \frac{5}{3} \] \[ f(3) = 32 \]