Question

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \[ \int_0^{\frac{\pi}{4}} \left( \sin(x) f(x) + \cos(x) \int_0^x f(t) \, dt \right) \, dx = \sqrt{2}. \] Then, the value of \[ \int_0^{\frac{\pi}{4}} f(x) \, dx \] is equal to ............... (rounded off to two decimal places).

Hint:

When dealing with nested integrals, use integration by parts or the Leibniz rule to simplify the problem.

Answer 1

Step 1: Break down the given equation.
The integral can be written as: \[ \int_0^{\frac{\pi}{4}} \sin(x) f(x) \, dx + \int_0^{\frac{\pi}{4}} \cos(x) \left( \int_0^x f(t) \, dt \right) dx = \sqrt{2}. \] Step 2: Use integration by parts.
For the second term, apply the Leibniz rule for differentiating an integral: \[ \frac{d}{dx} \left( \int_0^x f(t) \, dt \right) = f(x). \] So, the second term becomes: \[ \int_0^{\frac{\pi}{4}} \cos(x) \left( \int_0^x f(t) \, dt \right) dx. \] This can be simplified by switching the order of integration: \[ \int_0^{\frac{\pi}{4}} \int_t^{\frac{\pi}{4}} \cos(x) \, dx \, f(t) \, dt. \] Step 3: Solve the integral.
After applying integration by parts and simplifying, we find that: \[ \int_0^{\frac{\pi}{4}} f(x) \, dx = 1. \] Final Answer: \[ \boxed{1}. \]