u(x) = 1 + (1/4) ∫ (cos x * u(t)) dt from 0 to π/2

Steps to Solve

1. Identifying the Given Function The function is defined as: $$ u(x) = 1 + \frac{1}{4} \int_{0}^{\frac{\pi}{2}} \cos x \cdot u(t) , dt $$ Here, $u(x)$ depends on the variable $x$ and involves an integral.

2. Simplifying the Integral Since the integral is independent of the variable $x$, we can factor $\cos x$ out: $$ u(x) = 1 + \frac{\cos x}{4} \int_{0}^{\frac{\pi}{2}} u(t) , dt $$

3. Defining a Constant Let: $$ C = \int_{0}^{\frac{\pi}{2}} u(t) , dt $$ Then, we can rewrite the equation: $$ u(x) = 1 + \frac{\cos x}{4} C $$

4. Finding C by Integration To find $C$, we can substitute back $u(t)$ into the integral: $$ C = \int_{0}^{\frac{\pi}{2}} \left( 1 + \frac{\cos t}{4} C \right) dt $$ Calculate the integral: $$ C = \int_{0}^{\frac{\pi}{2}} 1 , dt + \frac{C}{4} \int_{0}^{\frac{\pi}{2}} \cos t , dt $$

5. Calculating the definite integrals Calculate the first integral: $$ \int_{0}^{\frac{\pi}{2}} 1 , dt = \frac{\pi}{2} $$ Calculate the second integral: $$ \int_{0}^{\frac{\pi}{2}} \cos t , dt = [\sin t]_{0}^{\frac{\pi}{2}} = 1 $$

6. Solving the Equation for C Substituting the values back gives us: $$ C = \frac{\pi}{2} + \frac{C}{4} $$ Multiplying through by 4 to eliminate the fraction: $$ 4C = 2\pi + C $$ Rearranging the terms: $$ 3C = 2\pi $$ Thus, $$ C = \frac{2\pi}{3} $$

7. Final Substitution for u(x) Substituting $C$ back into $u(x)$: $$ u(x) = 1 + \frac{\cos x}{4} \cdot \frac{2\pi}{3} $$ Simplifying gives: $$ u(x) = 1 + \frac{\pi \cos x}{6} $$