How do you integrate #int xsinx# by integration by parts method?
If you are studying maths, then you should learn the formula for Integration By Parts (IBP), and practice how to use it:
I was taught to remember the less formal rule in word; "The integral of udv equals uv minus the integral of vdu". If you struggle to remember the rule, then it may help to see that it comes a s a direct consequence of integrating the Product Rule for differentiation.
Essentially we would like to identify one function that simplifies when differentiated, and identify one that simplifies when integrated (or is at least is integrable).
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To integrate ( \int x \sin(x) ) using integration by parts, follow these steps:

Choose ( u ) and ( dv ): Let ( u = x ) and ( dv = \sin(x) , dx ).

Calculate ( du ) and ( v ): Differentiate ( u ) to get ( du ), which is ( du = dx ), and integrate ( dv ) to get ( v ), which is ( v = \cos(x) ).

Apply the integration by parts formula: [ \int u , dv = uv  \int v , du ]

Substitute the values: [ \int x \sin(x) , dx = x \cos(x)  \int (\cos(x)) , dx ]

Simplify and integrate: [ \int x \sin(x) , dx = x \cos(x) + \int \cos(x) , dx ]

Integrate ( \cos(x) ) to get the final result: [ \int x \sin(x) , dx = x \cos(x) + \sin(x) + C ] where ( C ) is the constant of integration.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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