Question

(b) The value of \( \int x \sin x \, dx \) will be:

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

Hint:

When solving integrals using parts, carefully choose \( u \) and \( dv \) for easier differentiation and integration.

Answer 1

The Correct Option is A

Usingintegrationbyparts,let: \[ u=x\quad\text{and}\quaddv=\sinx\,dx. \] Then: \[ du=dx\quad\text{and}\quadv=-\cosx. \] Applyingtheformula\(\intu\,dv=uv-\intv\,du\): \[ \intx\sinx\,dx=-x\cosx+\int\cosx\,dx. \] \[ \intx\sinx\,dx=-x\cosx+\sinx+C. \]