Question

Trying to derive integral of tan(x) using integration by parts, and I'm getting a strange result..

Answer 1

What's been happening is this: I rewrite \[\int\limits_{a}^{b}\tan x. dx\] as \[\int\limits_{a}^{b}\csc x \sin x.dx\] then do integration-by-parts with u = csc x, and dv = sin x dx.

Answer 2

[oops, meant sec(x) above, not csc(x)...

but, so then I get \[uv - \int\limits_{a}^{b}v.du \rightarrow \sec x(-\cos x) - \int\limits_{a}^{b}(-\cos x)(\sec x \tan x).dx\]

Answer 3

but when I simplify, I get \[\int\limits_{a}^{b}\tan x.dx = -1 + \int\limits_{a}^{b}\tan x.dx\]

Answer 4

Finding the integral of tangent using u-sub is pretty easy, but by parts seems interesting.

Answer 5

I tried it also with u = sin x and dv = sec x, but it got pretty messy. I at least saw the ln(cos(x)) term show up, but it was buried in a bunch of other garbage.

Answer 6

Hmm, seems like parts goes to nowhere...

Answer 7

Yeah, u-sub is definitely the way to go for that one. :-)

Answer 8

yeah, doesn't seem like tangent is integrable by parts. Try this one with parts, it works out fairly nicely :)\[\int\limits \sec^3 x dx\]

Answer 9

I'm still not sure how I got that result. It seems like I must have made a mistake somewhere, because it eventually gave me 0 = -1. !!
I wonder if it's similar to trying to integrate x^-1 with the power rule.

Answer 10

is that secant-cubed?

Answer 11

(laptop screen is low-res..)

Answer 12

Yeah, secant cubed :)

Answer 13

ok, doesn't seem too bad..