Let for some function y=f(x), ∫0xtf(t)dt=x2f(x), x>0 and f(2)=3. Then f(6) is equal to
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Step-by-Step Solution
Step 1: Differentiate both sides
We are given an integral equation. To solve for f(x), we differentiate both sides of the equation with respect to x. On the left side, we use the Fundamental Theorem of Calculus. On the right side, we use the product rule for differentiation.
Step 2: Apply differentiation rules
Applying the Fundamental Theorem of Calculus to the left side, we get x f(x). For the right side, using the product rule dxd(uv)=u′v+uv′, where u=x2 and v=f(x), we get 2xf(x)+x2f′(x).
Step 3: Rearrange and separate variables
Now, we rearrange the equation to isolate f'(x) and f(x) terms. We move 2x f(x) to the left side, then divide by x2f(x) to separate the variables, preparing for integration.
Step 4: Integrate both sides
We integrate both sides of the separated equation. The integral of f(x)f′(x) is ln∣f(x)∣, and the integral of −x1 is −ln∣x∣. We add the constant of integration C. Since x>0, we can drop the absolute value for x.
Step 5: Solve for f(x)
Using logarithm properties, −ln∣x∣ can be written as ln∣x−1∣. Then, exponentiating both sides, we get f(x)=eln∣x−1∣+C=eln∣x−1∣eC. Let A=eC, which is a positive constant. So, f(x)=xA.
Step 6: Find the constant A
We are given the condition f(2)=3. We substitute x=2 into our derived function f(x)=xA and set it equal to 3 to find the value of the constant A.
Step 7: Calculate f(6)
Now that we have found A=6, our function is f(x)=x6. We can substitute x=6 into this function to find the value of f(6).