It's a bit like a mathematical treasure hunt, isn't it? You're presented with an equation that looks a little daunting at first glance: 6 + ∫[a to x] (f(t)/t²) dt = 2√x for all x > 0. The mission? To find a specific function, f(t), and a particular number, a, that make this statement true. It's the kind of problem that makes you lean in, a little puzzle waiting to be solved.
Let's break it down. We have an integral, a constant term, and a term involving the square root of x. The integral itself contains an unknown function f(t) and a constant lower limit a. The fact that this equation holds true for all x > 0 is a huge clue. It means the relationship is fundamental, not just a fluke for a single value.
When you encounter an equation like this, especially one involving an integral that's equated to a known function, a common strategy is to differentiate both sides with respect to x. This often helps to eliminate the integral and reveal more about the unknown function.
Let's try that. Differentiating 6 + ∫[a to x] (f(t)/t²) dt with respect to x gives us f(x)/x² (thanks to the Fundamental Theorem of Calculus). Differentiating the right side, 2√x, gives us 2 * (1/2) * x^(-1/2), which simplifies to 1/√x.
So, we now have the equation: f(x)/x² = 1/√x.
From here, it's a straightforward step to isolate f(x). Multiply both sides by x²:
f(x) = x² * (1/√x)
f(x) = x² * x^(-1/2)
f(x) = x^(2 - 1/2)
f(x) = x^(3/2)
So, we've found our function: f(x) = x^(3/2). That's one part of the puzzle solved!
Now, what about the number a? We need to plug our newfound function back into the original equation and see if we can determine a. The original equation is:
6 + ∫[a to x] (t^(3/2) / t²) dt = 2√x
Let's simplify the integrand first:
t^(3/2) / t² = t^(3/2 - 2) = t^(-1/2)
So the equation becomes:
6 + ∫[a to x] t^(-1/2) dt = 2√x
Now, let's evaluate the integral: The antiderivative of t^(-1/2) is t^(1/2) / (1/2), which is 2√t.
Applying the limits of integration:
∫[a to x] t^(-1/2) dt = [2√t] from a to x = 2√x - 2√a
Substituting this back into our equation:
6 + (2√x - 2√a) = 2√x
Now, we can simplify this equation. Notice that 2√x appears on both sides, so we can subtract it from both sides:
6 - 2√a = 0
Rearranging to solve for a:
6 = 2√a
3 = √a
Squaring both sides to find a:
a = 3²
a = 9
And there we have it! The function is f(x) = x^(3/2) and the number is a = 9. It's satisfying when the pieces of a mathematical puzzle just click into place, revealing a clear and elegant solution.
