Question

The value of \(\displaystyle\int x\cos x\,dx\) is:

• A. \(\cos x+x\sin x+c\)
• B. \(x\sin x+c\)
• C. \(x\cos x+c\)
• D. \(\sin x+x\cos x+c\)

Hint:

Integrate by parts with \(u=x\), \(dv=\cos x\,dx\).

Answer 1

The Correct Option is A

Step 1: Differentiating each option instead:
Differentiate option A: \(\dfrac{d}{dx}[\cos x+x\sin x+c]=-\sin x+\sin x+x\cos x=x\cos x\).

Step 2: Confirming the match:
This equals the original integrand exactly, so option A is the correct antiderivative.

Final Answer:
Differentiation check confirms \(\boxed{\cos x+x\sin x+c}\).