Question

Integrate\[\int \sec x\text dx\]

Answer 1

How

Answer 2

May be, we have to use this following Identity:

\[1 + \tan^2x = \sec^2x\]

Answer 3

multiply topc and bottom by \(secx+tanx\)

Answer 4

\[\int\sec x\cdot\text dx=\int\sqrt{1+\tan x}\cdot\text dx\]

Answer 5

\[\int secx dx\]\[=\int secx\times \frac{secx+tanx}{secx+tanx} dx\]\[=\int \frac{sec^2x+secxtanx}{secx+tanx} dx\]\[=\int \frac{1}{secx+tanx} d(secx+tanx)\]\[=\ln |secx+tanx|+C\]

Answer 6

I remember long time ago, someone told me that it can be further simplified, but I don't know how to do that and forget the simplified answer...

Answer 7

Or..
\[\int\limits\frac{secx}{1} *\frac{secx+tanx}{secx+tanx} dx => \int\limits\frac{secx(secx+tanx}{secx+tanx} dx \]
\[u=secx+tanx =>du = (secxtanx+sec^2x)dx => dx = \frac{du}{secxtanx+sec^{2}x}
\]
\[\int\limits\frac{\sec^{2}x+secxtanx}{u} *\frac{du}{\sec^{2}x+sectanx} => \int\limits\frac{1}{u} du\]

Answer 8

\[\int\sec x\cdot\text dx\]\[=\int \sec x \times\frac{\sec x+\tan x}{\sec x+\tan x}\text dx\]\[=\int \frac{\sec^2 x+\sec x\tan x}{\sec x+\tan x}\text dx\]

Answer 9

\[u={\sec x+\tan x}\text dx\]\[\text du=\tan x\sec x+\sec^2x\]\[\int \frac{\sec^2 x+\sec x\tan x}{\sec x+\tan x}\text dx\Longrightarrow\int\frac{\text du}{u}\]\[=\ln u+c\]\[=\ln|\sec x+\tan x|+c\]

Answer 10

Thank you @Mimi_x3 & @Callisto

Answer 11

u = secx + tanx
du = tanxsecx + sec^2x dx ....

Answer 12

oh yes i shouldn't forget the \(\text dx\)

Answer 13

Hmm... you've put dx after ''u=secx+tanx'' ...

Answer 14

ah its just on the wrong line

Answer 15

Yes...

Answer 16

should be:
\[\text{let } u={\sec x+\tan x}\]\[\text du=\left(\tan x\sec x+\sec^2x\right)\text dx\]

Answer 17

Yes!!!

Answer 18

cool well have another piece to the puzzle
http://openstudy.com/users/unklerhaukus#/updates/4fe68a72e4b02c91101b333e

Answer 19

Haven't learnt DE yet.. I'm sorry that I can't help!!!

Answer 20

just so you know...this...
http://math2.org/math/integrals/more/sec.htm

is the first thing that pops up when you do a google search for "integral of sec"

Answer 21

say YO! if you actually though of using that u-substitution on your own and not because you already saw it done... :)

Answer 22

my first though was a trig substitution like waterineyes suggested, this didn't work so i checked Wolfram which gave a useless response, so i asked the question to OS, which gave the best outcome

Answer 23

thought *