Question

how to integrate: x sec^2 (x/2) dx

Answer 1

\[\int\limits x \sec^2 \frac{x}{2} dx\]

Answer 2

Use integration by parts.

\[\int u.dv = uv - \int v.du\]

\[u = x\]\[du = dx\]
\[dv = \sec^2 ~x ~dx\]\[v = \tan~x\]

\[\therefore \int x~\sec^2~x~dx = uv - \int v.du\]\[~~~~~~~~~~~~~~~~~~~~~~~~~~= x~\tan~x - \int \tan x~ dx\]\[~~~~~~~~~~~~~~~~~~~~~~~~~~= x~\tan~x - \log ~\sec~x + C\]

Answer 3

I generalized it there. I think you can do the same with \(x' = \frac{x}{2}\) also. :)