Key Concepts and Formulas

• Integral Equations with Separable Kernels: An integral equation of the form $f(x) = g(x) + \int_a^b K(x,y) f(y) dy$, where the kernel $K(x,y)$ can be written as $h(x)k(y)$, can be simplified by evaluating the definite integral as a constant.
• Integration by Parts: The formula $\int u\,dv = uv - \int v\,du$ is used to integrate products of functions.
• Substitution Rule: Used to simplify integrals by changing the variable of integration.
• Trigonometric Identities: The double angle identity $\sin(2y) = 2\sin y\cos y$ can simplify integrals involving products of sine and cosine.

Step-by-Step Solution

Step 1: Simplify the Integral Equation The given integral equation is $f(x) = x + \int\limits_0^{\pi /2} {\sin x \cos y\,f(y)\,dy}$. Since $\sin x$ is independent of the integration variable $y$, it can be taken outside the integral: $f(x) = x + \sin x \int\limits_0^{\pi /2} {\cos y\,f(y)\,dy}$

Step 2: Introduce a Constant for the Integral Term The integral $\int\limits_0^{\pi /2} {\cos y\,f(y)\,dy}$ is a definite integral with constant limits. Therefore, its value will be a constant. Let's denote this constant by $K$: $K = \int\limits_0^{\pi /2} {\cos y\,f(y)\,dy}$ Substituting this back into the simplified equation gives us the form of $f(x)$: $f(x) = x + K\sin x \quad (*)$

Step 3: Set up an Equation to Solve for the Constant K To find the value of $K$, we substitute the expression for $f(x)$ from equation $(*)$ back into the definition of $K$. Since the integration variable in the definition of $K$ is $y$, we replace $x$ with $y$ in equation $(*)$: $f(y) = y + K\sin y$ Now substitute this $f(y)$ into the integral for $K$: $K = \int\limits_0^{\pi /2} {\cos y\,(y + K\sin y)\,dy}$ Distribute $\cos y$: $K = \int\limits_0^{\pi /2} {(y\cos y + K\sin y\cos y)\,dy}$ Split the integral into two parts: $K = \int\limits_0^{\pi /2} {y\cos y\,dy} + K\int\limits_0^{\pi /2} {\sin y\cos y\,dy} \quad (**)$

Step 4: Evaluate the First Integral: $\int\limits_0^{\pi /2} {y\cos y\,dy}$ We use integration by parts with $u = y$ and $dv = \cos y\,dy$. Then $du = dy$ and $v = \sin y$. $\int\limits_0^{\pi /2} {y\cos y\,dy} = [y\sin y]_0^{\pi /2} - \int\limits_0^{\pi /2} {\sin y\,dy}$ $= \left(\frac{\pi}{2}\sin\frac{\pi}{2} - 0\sin 0\right) - [-\cos y]_0^{\pi /2}$ $= \left(\frac{\pi}{2} \cdot 1 - 0\right) - (-\cos\frac{\pi}{2} - (-\cos 0))$ $= \frac{\pi}{2} - (0 - (-1))$ $= \frac{\pi}{2} - 1$

Step 5: Evaluate the Second Integral: $\int\limits_0^{\pi /2} {\sin y\cos y\,dy}$ We can use the substitution $t = \sin y$, so $dt = \cos y\,dy$. The limits change from $y=0$ to $t=\sin 0 = 0$, and from $y=\pi/2$ to $t=\sin(\pi/2) = 1$. $\int\limits_0^{\pi /2} {\sin y\cos y\,dy} = \int\limits_0^1 {t\,dt}$ $= \left[\frac{t^2}{2}\right]_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}$ Alternatively, using the identity $\sin y\cos y = \frac{1}{2}\sin(2y)$: $\int\limits_0^{\pi /2} {\frac{1}{2}\sin(2y)\,dy} = \frac{1}{2}\left[-\frac{\cos(2y)}{2}\right]_0^{\pi /2} = -\frac{1}{4}[\cos(\pi) - \cos(0)] = -\frac{1}{4}[-1 - 1] = -\frac{1}{4}[-2] = \frac{1}{2}$

Step 6: Solve for K Substitute the evaluated integrals back into equation $(**)$: $K = \left(\frac{\pi}{2} - 1\right) + K\left(\frac{1}{2}\right)$ Rearrange to solve for $K$: $K - \frac{1}{2}K = \frac{\pi}{2} - 1$ $\frac{1}{2}K = \frac{\pi}{2} - 1$ $K = 2\left(\frac{\pi}{2} - 1\right)$ $K = \pi - 2$

Step 7: Write the Final Expression for f(x) Substitute the value of $K$ back into equation $(*)$: $f(x) = x + (\pi - 2)\sin x$

Common Mistakes & Tips

• Treating x as a constant inside the integral: Remember that the integral is with respect to $y$, so any term solely dependent on $x$ (like $\sin x$) can be moved outside the integral.
• Errors in integration by parts: Carefully identify $u$ and $dv$ and ensure correct application of the formula, especially with the limits.
• Forgetting to change limits in substitution: When using substitution for definite integrals, always update the integration limits to match the new variable.

Summary

The problem is solved by recognizing that the integral term in the given equation is a constant. This allows us to express $f(x)$ in a simpler form, $f(x) = x + K\sin x$. By substituting this form back into the definition of $K$, we obtain a linear equation for $K$, which can be solved by evaluating two separate definite integrals using integration by parts and substitution. Once $K$ is found, we substitute it back into the expression for $f(x)$ to get the final solution.

The final answer is $\boxed{x + (\pi - 2)\sin x}$.