We can solve this differential equation by recognizing it as an exact derivative. Dividing by x and multiplying by x³ (which is an integrating factor), we get 5x^4 f'(x) + 4x³ f(x) = 5x^7, which simplifies to (d)/(dx) (x^4 f(x)) = x^7. Integrating both sides with respect to x gives x^4 f(x) = (x^8)/(8) + C.

We can solve this differential equation by recognizing it as an exact derivative. Dividing by x and multiplying by x³ (which is an integrating factor), we get 5x^4 f'(x) + 4x³ f(x) = 5x^7, which simplifies to (d)/(dx) (x^4 f(x)) = x^7. Integrating both sides with respect to x gives x^4 f(x) = (x^8)/(8) + C.

To find the constant C, we use the original integral equation. Substituting x=1 into the original equation gives ∫_1^1 f(t) dt = 5(1)f(1) - 1^5 - 9. The left side is 0. This simplifies to 0 = 5f(1) - 10, so f(1) = 2. Now, substitute x=1 and f(1)=2 into x^4 f(x) = (x^8)/(8) + C to find C. This gives 1^4 (2) = (1^8)/(8) + C, so 2 = (1)/(8) + C, which means C = (15)/(8).

To find the constant C, we use the original integral equation. Substituting x=1 into the original equation gives ∫_1^1 f(t) dt = 5(1)f(1) - 1^5 - 9. The left side is 0. This simplifies to 0 = 5f(1) - 10, so f(1) = 2. Now, substitute x=1 and f(1)=2 into x^4 f(x) = (x^8)/(8) + C to find C. This gives 1^4 (2) = (1^8)/(8) + C, so 2 = (1)/(8) + C, which means C = (15)/(8).

With C = (15)/(8), the function becomes x^4 f(x) = (x^8)/(8) + (15)/(8). Dividing by x^4, we get the explicit form of f(x).

Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $\int_1^x f(t)\,dt = 5x f(x) - x^5 - 9$ for all $x \geq 1$ , then the value of $f(3)$ is:

Answer: $\frac{274}{27}$

Step-by-step solution

Step 1: Differentiate both sides

To solve this problem, we need to find the function $f(x)$ . The given equation involves an integral, so we differentiate both sides with respect to $x$ . This will eliminate the integral on the left side using the Fundamental Theorem of Calculus.

Step 2: Apply Fundamental Theorem of Calculus and Product Rule

Applying the Fundamental Theorem of Calculus, the derivative of the left side is $f(x)$ . On the right side, we use the product rule for $5x f(x)$ and the power rule for $x^5$ . The derivative of the constant 9 is 0.

Step 3: Rearrange the equation into a linear first-order differential equation

Now, we rearrange the equation to group terms involving $f(x)$ and $f'(x)$ . This results in a first-order linear differential equation.

Step 4: Solve the differential equation

We can solve this differential equation by recognizing it as an exact derivative. Dividing by $x$ and multiplying by $x^3$ (which is an integrating factor), we get $5x^4 f'(x) + 4x^3 f(x) = 5x^7$ , which simplifies to $\frac{d}{dx} (x^4 f(x)) = x^7$ . Integrating both sides with respect to $x$ gives $x^4 f(x) = \frac{x^8}{8} + C$ .

Step 5: Find the constant of integration C

To find the constant $C$ , we use the original integral equation. Substituting $x=1$ into the original equation gives $\int_1^1 f(t)\,dt = 5(1)f(1) - 1^5 - 9$ . The left side is 0. This simplifies to $0 = 5f(1) - 10$ , so $f(1) = 2$ . Now, substitute $x=1$ and $f(1)=2$ into $x^4 f(x) = \frac{x^8}{8} + C$ to find $C$ . This gives $1^4 (2) = \frac{1^8}{8} + C$ , so $2 = \frac{1}{8} + C$ , which means $C = \frac{15}{8}$ .

Step 6: Determine f(x)

With $C = \frac{15}{8}$ , the function becomes $x^4 f(x) = \frac{x^8}{8} + \frac{15}{8}$ . Dividing by $x^4$ , we get the explicit form of $f(x)$ .

Step 7: Calculate f(3)

Finally, substitute $x=3$ into the expression for $f(x)$ to find the value of $f(3)$ .

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Let f:[1,∞)→[2,∞)f : [1, \infty) \to [2, \infty)f:[1,∞)→[2,∞) be a differentiable function. If ∫1xf(t) dt=5xf(x)−x5−9\int_1^x f(t)\,dt = 5x f(x) - x^5 - 9∫1x​f(t)dt=5xf(x)−x5−9 for all x≥1x \geq 1x≥1, then the value of f(3)f(3)f(3) is: