Question

find the derivative of ( f(x)=xsin(x)+cos(x)+c ) to complete the following integration formula: ( intsquare dx=xsin(x)+cos(x)+c )

Explanation:

Step1: Apply product rule

The product rule states that \((uv)^\prime = u^\prime v+uv^\prime\). For \(u = x\) and \(v=\sin(x)\), \(u^\prime=1\) and \(v^\prime=\cos(x)\). So \((x\sin(x))^\prime=1\times\sin(x)+x\cos(x)=\sin(x)+x\cos(x)\).

Step2: Differentiate \(\cos(x)\) and \(C\)

The derivative of \(\cos(x)\) is \(-\sin(x)\), and the derivative of a constant \(C\) is \(0\).

Step3: Combine the derivatives

\(f^\prime(x)=(\sin(x)+x\cos(x))-\sin(x)+0\)

Answer:

\(x\cos(x)\)