Let for some function y=f(x), ∫₀ˣ t f(t) d t=x² f(x), x>0 and f(2)=3. Then f(6) is equal to

Solution

Maths sir · 4 steps

An integral with variable upper limit is begging to be differentiated, so differentiate both sides.

1. Differentiate both sides. By the Fundamental Theorem of Calculus, the derivative of ∫₀ˣ t f(t) dt is x f(x). The right side x² f(x) needs the product rule.

   x f(x) = 2x f(x) + x² f'(x)

2. Separate variables. This gives x² f'(x) = -x f(x), so f'(x)/f(x) = -1/x for x > 0. Integrating, ln f(x) = -ln x + C, so f(x) = c/x.
3. Use f(2) = 3. c/2 = 3 gives c = 6, so f(x) = 6/x. Check: ∫₀ˣ t · 6/t dt = 6x and x² · 6/x = 6x, so the condition holds.
4. So. f(6) = 6/6 = 1, which is option (a).

The idea to keep. When the unknown function sits inside an integral with x as a limit, differentiate both sides. The integral equation turns into a differential equation you can separate.

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