Question

If \(\displaystyle \int_{9}^{x}\frac{f(y)}{y^2}\,dy=2\sqrt{x}-6\), then \(f(x)=\)

• A. \(\sqrt{x}\)
• B. \(x\sqrt{x}\)
• C. \(x^2\sqrt{x}\)
• D. \(x+\sqrt{x}\)

Hint:

If \[ \int_a^x g(y)\,dy=F(x), \] then by the Fundamental Theorem of Calculus, \[ g(x)=F'(x). \] This is very useful for finding unknown functions inside definite integrals.

Answer 1

The Correct Option is B

Step 1: Apply Leibniz rule for differentiating under the integral.
d/dx [∫₉ˣ f(y)/y² dy] = f(x)/x² by the Fundamental Theorem. Right side derivative: d/dx[2x^(1/2) – 6] = x^(–1/2).

Step 2: Equate and solve for f(x).
f(x)/x² = 1/√x → f(x) = x²/√x = x^(3/2).

Step 3: Final Answer:
x√x.