How can you use the tabular method to integrate xcos(x^2)

[lioric]

Homework Statement

This is not a homework but since asked me I'm posting it here. I know how to intergrate by parts and can do this using formula
But I'd like to do this using the tabular method

Question
Integrate xcos(x^2) using tabular method

Homework Equations

The table with derivative on one volume and integral on the other column

The Attempt at a Solution

I know we can differentiate the one which is harder to intergrate so I integrated the x and differentiated cos(x^2)

So I get -2sin(x^2) for the derivative column and (x^2)/2 on the intergral side
When I multiply the results as said in tabular method and integrate the last portion, I get

(X^2)/2 * cos(x^2) -2 integrated by sin(x^2) * (x^2)/2

 

[NFuller]

I know how to intergrate by parts and can do this using formula
But I'd like to do this using the tabular method

The tabular method is integration by parts. It is just organizing the steps into a table. If you understand the formula than there is no need to make a table.

I know we can differentiate the one which is harder to intergrate so I integrated the x and differentiated cos(x^2)

So I get -2sin(x^2) for the derivative column and (x^2)/2 on the intergral side
When I multiply the results as said in tabular method and integrate the last portion, I get

(X^2)/2 * cos(x^2) -2 integrated by sin(x^2) * (x^2)/2

The first derivative is $-2x\text{sin}(x^{2})$ not $-2\text{sin}(x^{2})$. I don't think this is a good method to solve this problem though. This integral is relatively trivial using the u-substitution $u=x^{2}$.