Integral problem, faculty degree

[mikihami666]

Does anyone know how to solve this integral:
Divide[Exp[ArcTan[x]],Power[\(40)1+Power[x,2]\(41),Divide[3,2]]] dx
The picture provided shows only the first part, partial integration but the second half when teacher uses "I" doesnt even make sence, can someone explain?

 

[Dr.Peterson]

Does anyone know how to solve this integral:
Divide[Exp[ArcTan[x]],Power[\(40)1+Power[x,2]\(41),Divide[3,2]]] dx
The picture provided shows only the first part, partial integration but the second half when teacher uses "I" doesnt even make sence, can someone explain?
View attachment 33663

This is an example of a technique sometimes described as "looping", where applying integration by parts twice brings you back to the original integral. By giving that a name (commonly I), you end up with an equation to solve.

Example 8 here is a simpler example of the same technique, where the integral is simple enough that they didn't have to call it I:

Calculus II - Integration by Parts

In this section we will be looking at Integration by Parts. Of all the techniques we’ll be looking at in this class this is the technique that students are most likely to run into down the road in other classes. We also give a derivation of the integration by parts formula.

tutorial.math.lamar.edu

Here is another explanation of the same thing:

Integration by Parts Problems where You Go around in Circles - dummies

Sometimes if you use integration by parts twice, you get back to where you started from — which, unlike getting lost, is not a waste of time. First, use the LIA

www.dummies.com

 

[Steven G]

Your teacher just gave that integral a name, namely I.
Why did your teacher do that? Just to make notation easier as your teacher wanted to combine the integrals that are the same. If you choose, you can not use I and write out the entire integral.