Question

integral of e^x(sec(e^x))

Answer 1

i know that the integral of \[e^{x}\sec e ^{x} \]
= \[\ln \left| \sec x \tan x \right| + c\]
but what is the process

Answer 2

substitution
also I think you are missing a + sign and some e^x in that answer

Answer 3

\[
\int e^x\sec(e^x)~dx=\int \sec(e^x)~d(e^x) = \int \sec u~du
\]

Answer 4

-_- but where does the e^x go?

Answer 5

let u=e^x
then du=e^x dx

Answer 6

Well, \(e^x=u\).

Answer 7

no i mean, from the answer

Answer 8

the answer is incorrect

Answer 9

Well, \(x=\ln(u)\).

Answer 10

thats the answer my teacher told me to get to -_-

Answer 11

\[\int\limits_{}^{}\sec(u) du=\ln|\sec(u)+\tan(u)|+C\]

Answer 12

it is usually what we program to remember

Answer 13

but we can derive this finding

Answer 14

\[\int\limits_{}^{}\sec(u) \frac{\sec(u)+\tan(u)}{\sec(u)+\tan(u)} du \]
you do a substitution here

Answer 15

let v=sec(u)+tan(u)
then dv=sec(u)(tan(u)+sec(u)) du

Answer 16

\[\int\limits_{}^{}\frac{dv}{v}\]
do you know what that equals?

Answer 17

nope, im confused

Answer 18

are you suppose to remember \[\int\limits_{}^{}\sec(x) dx=\ln|\sec(x)+\tan(x)|+C \]
or have you not seen this integral at all?

Answer 19

ive seen it online, but i havent seen it prior to today

Answer 20

usually you are allowed to remember this

Answer 21

so back to our integral
\[\int\limits_{}^{}e^x \sec(e^x) dx \\ \text{ Let } u=e^x \\ \text{ Then } du=e^x dx \\ \int\limits_{}^{} \sec(e^x) e^x dx=\int\limits_{}^{}\sec(u) du\]

Answer 22

so you can use that here

Answer 23

where u is e^x

Answer 24

i understand that, and i see what u mean by the answer is wrong, but i still need to find some way of getting to the answer -_-

Answer 25

I gave you a way of getting to the answer

Answer 26

ln|sec(u)+tan(u)|+C
replace u with e^x

Answer 27

\[\int\limits\limits_{}^{}e^x \sec(e^x) dx \\ \text{ Let } u=e^x \\ \text{ Then } du=e^x dx \\ \int\limits\limits_{}^{} \sec(e^x) e^x dx=\int\limits\limits_{}^{}\sec(u) du =\ln|\sec(u)+\tan(u)|+C \\ =\ln|\sec(e^x)+\tan(e^x)|+C\]

Answer 28

i get that, but how do i get from there to the answer my teacher gave me? is it not possible?

Answer 29

It's not impossible because it is not true.

Answer 30

lol make the correct answer wrong is all

Answer 31

\[\text{ Checking your teacher's answer } \\ \frac{d}{dx}\ln|\sec(x)\tan(x)| \\ \frac{(\sec(x)\tan(x))'}{\sec(x)\tan(x)}=\frac{\sec(x)\tan(x) \cdot \tan(x)+\sec(x) \cdot \sec^2(x)}{\sec(x)\tan(x)} =\tan(x)+\frac{\sec^2(x)}{\tan(x)} \\ =\tan(x)+\frac{\frac{1}{\cos^2(x)}}{\frac{\sin(x)}{\cos(x)}} =\tan(x)+\frac{1}{\sin(x)\cos(x)}\]
Don't see how this will magically give us e^x somewhere

Answer 32

also my lines got cut off

Answer 33

thx, i guess i will tell my teacher that his answer is wrong -_-

Answer 34

i guess getting the answer wasn't the thing that was confusing you
it was his answer that was confusing you

Answer 35

ya -_-

Answer 36

if he had \[\int\limits_{}^{}\frac{\sec(x)\tan^2(x)+\sec^3(x)}{\sec(x)\tan(x)} dx \text{ or if we had } \int\limits_{}^{}\frac{\tan^2(x)+\sec^2(x)}{\tan(x)} dx\]
then your teacher would have been right
just trying to think backwards from is solution

Answer 37

his solution*