Let f be a differentiable function such that x²f(x) - x = 4∫₀ˣ tf(t)dt, f(1) = 2/3. Then 18f(3) is equal to :

Solution

Maths sir · 4 steps

Differentiate once to remove the integral, and you are left with a linear equation in f.

1. Remove the integral. Differentiate both sides with respect to x. The right side becomes 4x f(x), so 2x f + x² f' - 1 = 4x f, which gives x² f' - 2x f = 1.
2. Integrating factor. Divide by x² to get f' - 2/x f = 1/x². The integrating factor is e^(-∫ 2/xdx) = 1/x².

   d/dx(f/x²) = 1/x⁴ ⇒ f/x² = -1/3x³ + C

3. Find C. Put x = 1: 2/3 = -1/3 + C, so C = 1. Hence f(x) = x² - 1/3x.
4. So. f(3) = 9 - 1/9 = 80/9, and 18f(3) = 18 × 80/9 = 160, option (a).

The idea to keep. When f sits inside an integral with variable limit, differentiate first. The integral disappears and you usually get a linear equation you can solve with an integrating factor.

Answer from the NTA final answer key. Solution by PrepWiser.