Integration by Parts

Why not? Line 6 is blatantly wrong (and I'm ashamed for it) but line 5? Technically int(a+b)dx = int(a)dx + int(b)dx, or am I forgetting something and making pomegranate up on the fly?

 Spoiler

Also, what the hell @ the last line, I totally turned a multiplication into an addition. Serves me right for doing Math without pencil and paper at midnight.

If anything, the last 3 lines are wrong. But until that point, I'm fairly sure I got it right.

 answer

first red part: xln(x^2 - 1)
second red part: -2x
blue part: -2*int(1/(x^2-1))dx = -2 * (ln(x-1) - ln(x+1))/2 = -ln(x-1) + ln(x+1)

And if I'm not hallucinating again, int (1/x^2-1)dx should be one of the hyperbolic integrands or its inverse. Nobody remembers that crap anyway, so I'd stick with integration by parts.

oh my bad, line 5 is fine, didn't know where that separation came from.

arctan(x) = int( 1/(x^2 + 1) )
...from memory
...took calc like 5 years ago
...close to math masters

dont need trig for partial fractions