Question

If \( f(x) \) is continuous and \( \int_0^9 f(x) \, dx = 4 \), then the value of the integral \( \int_0^3 x \cdot f(x^2) \, dx \) is: (a) 2

• A. 2
• B. 18
• C. 16
• D. 4

Hint:

When dealing with integrals involving transformations such as \( x^2 \), always use substitution to simplify the integral into a standard form.

Answer 1

The Correct Option is A

We simplify the given integral using substitution. Let: \[ t = x^2 \quad \Rightarrow \quad dt = 2x \, dx. \] Rewriting the integral in terms of \( t \): \[ I = \int_0^3 x \cdot f(x^2) \, dx. \] Using substitution: \[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x}. \] Thus, the integral becomes: \[ I = \int_0^3 x \cdot f(x^2) \, dx = \frac{1}{2} \int_0^9 f(t) \, dt. \] Given that: \[ \int_0^9 f(t) \, dt = 4, \] we substitute: \[ I = \frac{1}{2} \times 4 = 2. \]