Integrate x * sin(x) with respect to x by using integration by parts

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.

JW
Answered by Jamie W. Maths tutor

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