Question

antiderivatives: problem 3
(1 point)
find the derivative of ( f(x)=x sin (x)+cos (x)+c ) to complete the following integration formula:
( int square d x=x sin (x)+cos (x)+c )

Explanation:

Step1: Differentiate \(x\sin(x)\)

Use the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = x\), \(u^\prime=1\), \(v=\sin(x)\), \(v^\prime=\cos(x)\). So \((x\sin(x))^\prime=\sin(x)+x\cos(x)\).

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

The derivative of \(\cos(x)\) is \(-\sin(x)\).

Step3: Differentiate the constant \(C\)

The derivative of a constant \(C\) is \(0\).

Step4: Combine the derivatives

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

Simplify the expression: \(\sin(x)+x\cos(x)-\sin(x)=x\cos(x)\)

Answer:

\(x\cos(x)\)