Question

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

• A. \(x+\frac23(\pi-2)\sin x\)
• B. \(x+(\pi+2)\sin x\)
• C. \(x+\frac{\pi}{2}\sin x\)
• D. \(x+(\pi-2)\sin x\)

Hint:

The integral is a constant times \(\sin x\): write \(f(x)=x+C\sin x\) with \(C=\int_0^{\pi/2}\cos y\,f(y)\,dy\) and solve for \(C\).

Answer 1

The Correct Option is D

Since \(\sin x\) does not depend on \(y\): \(f(x)=x+C\sin x\), where \(C=\int_0^{\pi/2}\cos y\,f(y)\,dy\).
Substitute \(f(y)=y+C\sin y\): \(C=\int_0^{\pi/2}y\cos y\,dy+C\int_0^{\pi/2}\sin y\cos y\,dy\).
\(\int_0^{\pi/2}y\cos y\,dy=[y\sin y+\cos y]_0^{\pi/2}=\frac{\pi}{2}-1\); \(\int_0^{\pi/2}\sin y\cos y\,dy=\frac12\).
\(C=\frac{\pi}{2}-1+\frac C2\Rightarrow C=\pi-2\).
\(f(x)=x+(\pi-2)\sin x\).
Answer: (4) \(x+(\pi-2)\sin x\)