Question

Help with this integral:

I will write it out

Answer 1

\[\int\limits_{}^{} (\sec^3x+e ^{sinx})/(secx)\]

Answer 2

is there another way i can write this to make it easier to integrate

Answer 3

split it in two parts and you only have the right side to deal with

Answer 4

you mean like (sec^3x/secx_ and (e^sinx/ secx)

Answer 5

\[\int \frac{sec^3}{sec}+\int \frac{e^{sin}}{sec}\]
\[tan+\int \frac{e^{sin}}{sec}\]
yep

Answer 6

okay, now let me as you one more question

Answer 7

and 1/sec = cos which is the derivative of sin

Answer 8

okay, so then that (e^sinx/secx) becomes (e^sinx)(cosx)

Answer 9

\[\int cos\ e^{sin}=\int Du\ e^u\]

Answer 10

so what would be the u for that part: cosx right

Answer 11

if you need to rewrite it into us, then u=sin, and du = cos

Answer 12

at any rate it just ints up back into e^{sin}

Answer 13

okay,now lets say i had a different problem, where i picked e^sinx as u. And i differntiatied. And i got: (du)/(cosx* e^sinx). Could i rewrite that as secx* e^sinx

Answer 14

\[Dx(B^x)=B^x*ln(B)*x'\]
\[Dx(e^{sin})=e^{sin}*ln(e)*cos\]

Answer 15

because 1/cosx = secx

Answer 16

id personally forget about trying to make anything into a "u" and just work thru the mechanics that got you there :)\[\]
but if you want to U it and pick e^{sin}; then du = cos e^{sin}\[\]
which = e^{sin}/sec ; yes

Answer 17

\[\int \sec^2(x)+\cos(x)e^{\sin(x)}dx\] is what you have right?

Answer 18

yep

Answer 19

i havent done it yet, i was unsure so, i am working it through right now with the help i got

Answer 20

if i were to wrote out the rules of derivatives; I would always include the x' that pops out as a result of the chain rule

Answer 21

so, now i broke it up into two parts, (sec^x) and cosx*e^sinx. Now i will integrate the second one using substituiton for e^sinx

Answer 22

ok. @gremlin it is unlikely that you would make a substitution like
\[u=e^{\sin(x)}\] if you are trying to do a u-sub. this is itself a composite function so you want to make u = inside piece. in any case you might see just from your eyeballs that the anti - derivative of
\[\cos(x)e^{\sin(x)}\] is
\[e^{\sin(x)}\]

Answer 23

right

Answer 24

youre doing fine :)

Answer 25

or should i just say that u is sinx, the argument of e

Answer 26

u = sin(x) is the better choice

Answer 27

okay

Answer 28

right, because now i see that i would actually get the e^sinx back if i had picked it as u

Answer 29

now i got my final answer as being: tanx+e^sinx+c

Answer 30

the reason integration is harder than derivatives is simply because derivatives are very mechanical; they step down by established rules and guidleines; they are like having a dog as a pet.\[\]
integration is more catlike; it doesnt follow any general set of rules that will allow you to get to some known answer

Answer 31

that is correct

Answer 32

thank you for the help