# How do you integrate #int xcosx# by integration by parts method?

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To integrate ( \int x \cos(x) , dx ) using integration by parts, you apply the formula:

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

Let ( u = x ) and ( dv = \cos(x) , dx ). Then, differentiate ( u ) to find ( du ) and integrate ( dv ) to find ( v ).

[ du = dx ] [ v = \int \cos(x) , dx = \sin(x) ]

Now, apply the integration by parts formula:

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

The integral ( \int \sin(x) , dx ) can be easily evaluated, giving:

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

where ( C ) is the constant of integration.

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