The general formula for integration by parts to integrate something of the form u * v' is: u * v - (integral)[ (u' * v) dx ]. Thus we first need to write x * sin(x) in the form u * v'. Lets pick u = x and v' = sin(x), then we need to find u' and v. Differentiating u = x gives us u' = 1, while integrating v' = sin(x) gives us v = cos(x). Now we have: u = x, u' = 1, v = - cos(x), and v' = sin(x). All that's left to do is plug them into our general formula (outlined above). Therefore we have: - x * cos(x) - (integral)[(1 * -cos(x)) dx]. We're almost there, we just need to find (integral)[(1 * -cos(x)) dx]. This reduces to just integrating -cos(x), which equals -sin(x) + C. Putting that back into the formula leaves us with - x * cos(x) + sin(x) + C, which is the final answer (make sure that you dont forget the integration constant (+C) at the end). We can then check our answer by differentiating this to see if we can get back to x * sin(x). Differentiating - x * cos(x), we need to use the product rule, giving us -cos(x) + xsin(x). Differentiating sin(x) + C gives us cos(x) only. Combining these we find that the cos(x) terms cancel and we indeed are left with the xsin(x) that we started with.

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