Evaluate the following integral using integration by parts.
\[
\int x \sin x \cos x d x
\]
Let $u=x \sin x$. Use the integration by parts formula so that the new integral is simpler than the original one.
\[
\int x \sin x \cos x d x=\frac{1}{2} x^{2} \cos (x) \sin (x)-\int\left(\frac{1}{2} x^{2} \cos (2 x) d x\right.
\]
Evaluate the integral.
\[
\int x \sin x \cos x d x=\frac{1}{2}\left(-\frac{1}{2} x \cos (2 x)+\frac{1}{4} \sin (2 x)\right)+c
\]

Step 1 ：Let \(u=x \sin x\). Use the integration by parts formula so that the new integral is simpler than the original one.

Step 2 ：\[\int x \sin x \cos x d x=\frac{1}{2} x^{2} \cos (x) \sin (x)-\int\left(\frac{1}{2} x^{2} \cos (2 x) d x\right.\]

Step 3 ：Evaluate the integral.

Step 4 ：\[\int x \sin x \cos x d x=\frac{1}{2}\left(-\frac{1}{2} x \cos (2 x)+\frac{1}{4} \sin (2 x)\right)+c\]

Step 5 ：The integral of the function \(x \sin x \cos x\) can be solved using the integration by parts method. The first step is to identify the parts of the function to be used in the integration by parts formula. In this case, we can let \(u = x \sin x\) and \(dv = \cos x dx\). Then we need to find \(du\) and \(v\). After that, we substitute these into the integration by parts formula and simplify the resulting expression.

Step 6 ：Final Answer: \[\boxed{\frac{1}{4} x \sin^{2}(x) - \frac{1}{4} x \cos^{2}(x) + \frac{1}{4} \sin(x) \cos(x) + C}\]