Evaluate $\int x \sin x d x$

Step 1 ：First, we recognize that this is an integral that can be solved using integration by parts. The formula for integration by parts is \(\int u dv = uv - \int v du\).

Step 2 ：We let \(u = x\) and \(dv = \sin x dx\). Then we need to find \(du\) and \(v\).

Step 3 ：Differentiating \(u = x\) with respect to \(x\), we get \(du = dx\).

Step 4 ：Integrating \(dv = \sin x dx\) with respect to \(x\), we get \(v = -\cos x\).

Step 5 ：Substituting \(u\), \(v\), \(du\), and \(dv\) into the integration by parts formula, we get \(\int x \sin x dx = x(-\cos x) - \int -\cos x dx\).

Step 6 ：Solving the integral \(\int -\cos x dx\), we get \(-\sin x\).

Step 7 ：Substituting this back into our equation, we get \(\int x \sin x dx = x(-\cos x) - (-\sin x)\).

Step 8 ：Simplifying this, we get \(\int x \sin x dx = -x\cos x + \sin x\).

Step 9 ：Therefore, the solution to the integral \(\int x \sin x dx\) is \(\boxed{-x\cos x + \sin x}\).