Let us have an integral (In)
$$\int x \cos(x) \, dx $$
We get 8 combinations for u and dv, in order to integrate the given integral by parts, as follows
1- u=1, dv=x cosx dx
2- u=x cosx, dv= dx
3- u=x cosx dx, dv=1
4- u=x dx, dv= cosx
5- u= cosx dx, dv= x
6- u=x, dv=cosx dx
7- u= cosx, dv= x dx
8- u= dx, dv= x cosx
But according to my book, we have only 4 of these combinations valid, which are 1, 2, 6 and 7.
Idk what is the reason behind eliminitating 3, 4, 5 and 8.
what is the reason behind eliminating the combination 3, 4, 5 and 8? Why are these combinations not valid for 'integration by parts'
in this case you can set $u=x$ and $v'=\cos(x)$ then you will get $$u'=1$$ and $$v=\sin(x)$$ and our integral will be $$x\sin(x)-\int \sin(x)dx$$