(1) indicates that the original function of v' is clear at a glance, for example, v' is sinx, e x and so on.

But there is another usage of partial integral, which will be used when finding ∫secxdtanx.

∫secxdtanx

=∫secx*(tan^2x+ 1)dx

=∫secxdx+∫(secx*tanx)*tanxdx

=∫secxdx+∫tanxdsecx

Use partial integral at this time.

=∫secxdx+tanxsecx-∫secxdtanx

The reason for this is that you can get an equation.

∫secxdtanx =∫secxdx+tanxsecx-∫secxdtanx

2∫secxdtanx=∫secxdx+tanxsecx

In fact, both of them are difficult to integrate, but in fact, they don't need to be calculated. Just use partial integral equations to simplify the problem.