What is the result of integrating xsin(x) dx using integration by parts?

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To determine the result of integrating ( x \sin(x) , dx ) using integration by parts, we apply the integration by parts formula:

[

\int u , dv = uv - \int v , du

]

In this case, we can choose ( u = x ) and ( dv = \sin(x) , dx ). Then, we compute ( du ) and ( v ):

Now, substituting these into the integration by parts formula:

[

\int x \sin(x) , dx = x (-\cos(x)) - \int (-\cos(x)) , dx

]

This simplifies to:

[

-x \cos(x) + \int \cos(x) , dx

]

Next, we integrate ( \cos(x) ):

[

\int \cos(x) , dx = \sin(x)

]

So, putting it all together, we have:

[

\int x \sin(x) , dx = -x \cos(x) + \sin(x) + C

]

Since the