Question

Evaluate \( \int x \cos x \, dx \):

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

Hint:

When performing integration by parts, remember the formula \( \int u \, dv = uv - \int v \, du \).

Answer 1

The Correct Option is D

To solve \( \int x \cos x \, dx \), we apply integration by parts. 
Let: \[ u = x \quad {and} \quad dv = \cos x \, dx \] Then: \[ du = dx \quad {and} \quad v = \sin x \] Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get: \[ \int x \cos x \, dx = x \sin x - \int \sin x \, dx = x \sin x + \cos x + C \]