I was thinking about this integral

and it's simple to calculate when n is small, but when n is big it's really time-consuming, so I thought I could be smart and rewrite it like this:

and then integrate it. I thought it would make things easier as I figured that

.

Now I thought that I could simply do this

which would mean that

.

I realized that this is dead wrong as soon as I thought of setting a to zero, but I don't know where my mistake is. Could someone please explain why this doesn't work?