Let 5 f(x)+4 f(1/x)=1/x+3, x > 0. Then 18 ∫1² f(x) d x is equal to :

Solution

Maths sir · 4 steps

Swap x with 1/x to get a second equation, then solve the pair for f(x) like simultaneous equations.

1. Swap x and 1/x. Replace x by 1/x in the given equation. That gives 5 f(1/x) + 4 f(x) = x + 3. Now you have two linear equations in the two unknowns f(x) and f(1/x).
2. Eliminate f(1/x). Multiply the original equation by 5 and the new one by 4. Then subtract: 25 f(x) - 16 f(x) = 5/x + 15 - 4x - 12, so 9 f(x) = 5/x - 4x + 3.

   18∫₁² f(x) dx = 2∫₁²(5/x - 4x + 3)dx

3. Integrate. ∫₁² 5/x dx = 5 logₑ 2, ∫₁² 4x dx = 2(4 - 1) = 6 and ∫₁² 3 dx = 3. So the bracket is 5 logₑ 2 - 6 + 3 = 5 logₑ 2 - 3.
4. So. 18∫₁² f(x) dx = 2(5 logₑ 2 - 3) = 10 logₑ 2 - 6, which is option (c).

The idea to keep. Whenever f(x) and f(1/x) appear together, substitute x to 1/x to get a partner equation and eliminate. The same trick works for f(x) and f(−x), or f(x) and f(1−x).

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