The function f(x), that satisfies the condition f(x)=x+_0^/2 x y,f(y),dy, is

Question

The function $f(x)$, that satisfies the condition $f(x)=x+\displaystyle\int_0^{\pi/2}\sin x\cdot\cos y\,f(y)\,dy$, is

Accepted Answer

$$f(x) = x + \int\limits_0^{\pi /2} {\sin x\cos y\,f(y)\,dy} $$

$$f(x) = x + sinx\underbrace {\int_0^{\pi /2} {\cos y\,f(y)\,dy} }_K$$

$$ \Rightarrow f(x) = x + K\sin x$$

$$ \Rightarrow f(y) = y + K\sin y$$

Now, $$K = \int_0^{\pi /2} {\mathop {y\cos y\,dy}\limits_{Apply\,IBP} }  + K\int_0^{\pi /2} {\mathop {\cos y\sin y\,dy}\limits_{Put\,\sin y = t} } $$

$$K = \left( {y\sin y} \right)_0^{\pi /2} - \int_0^{\pi /2} {\sin y dy + K\int_0^1 {t\,dt} } $$

$$ \Rightarrow K = {\pi  \over 2} - 1 + K\left( {{1 \over 2}} \right)$$

$$ \Rightarrow K = \pi  - 2$$

So, $$f(x) = x + (\pi  - 2)\sin x$$

Option (d)

Solutions · 1