Which integration technique was used here?

[duarthiago]

I've just found the following example on Piskunov:

\int \frac{t^2 dt}{(t^2 + 2)^2}\\= \frac{1}{2}\int \frac{t d(t^2 +2)}{(t^2 + 2)^2}\\=-\frac{1}{2}\int t d(\frac{1}{t^2 + 2})\\=-\frac{1}{2}\frac{t}{t^2 + 2}+\int \frac{dt}{t^2 + 2}\\=-\frac{t}{2(t^2 + 2)} + \frac{1}{2 \sqrt{2}}arctan\frac{t}{\sqrt{2}}

What technique is this? Apparently there is a substitution at the beginning, but I can't figure out what happens from second line onwards.

 

[Mark44]

I've just found the following example on Piskunov:

\int \frac{t^2 dt}{(t^2 + 2)^2}\\= \frac{1}{2}\int \frac{t d(t^2 +2)}{(t^2 + 2)^2}\\=-\frac{1}{2}\int t d(\frac{1}{t^2 + 2})\\=-\frac{1}{2}\frac{t}{t^2 + 2}+\int \frac{dt}{t^2 + 2}\\=-\frac{t}{2(t^2 + 2)} + \frac{1}{2 \sqrt{2}}arctan\frac{t}{\sqrt{2}}

What technique is this? Apparently there is a substitution at the beginning, but I can't figure out what happens from second line onwards.

The substitution was w = t² + 2, so dw = 2tdt
In later steps it appears that they did integration by parts, with u = t and dv = dt/(t² + 2).

 

[duarthiago]

The substitution was w = t² + 2, so dw = 2tdt
In later steps it appears that they did integration by parts, with u = t and dv = dt/(t² + 2).

Of course! I didn't even try to see an integration by parts there. Thank you!