Question

HELP!!! How do you integrate e^x/1+e^2x*dx

Answer 1

can you type it using latex please (the equation tool)?

Answer 2

Im so sorry I don't know what latex is?

Answer 3

I typed it into Wolfram before, and got the final answer, but I dont know the steps in between

Answer 4

Using that equation thing at the bottom.

\[\int{e^x \over {1 + e^{2x}}} dx\]

does it look like that?

Answer 5

yes!

Answer 6

Oh, I see the equation tool, sorry

Answer 7

e^x = u

du = e^x dx

u^2 = e^2x

\[\int\limits {1 \over 1 + u^2} du\]

now letting u = tan k

du = sec^2 k dk

so it becomes:

\[\int\limits {\sec^2 k \over 1 + \tan^2k} dk\]

and that equals (because sec^2 k = tan^2 k + 1)

\[\int\limits {\sec^2 k \over \sec^2 k} dk = k + C\]

but u = tan k

k = arctan u

so it's arctan u + C

but u = e^x

so it's

arctan e^x + C

you could have directly made the substitution e^x = tan u, but this makes one understand it better

Answer 8

thanks soooo much, let me have a minute to digest it now.... lol

Answer 9

I don't understand this part: u^2 = e^2x

Answer 10

oops, yes I do!

Answer 11

haha :) cool

Answer 12

Last question, now how to I evaluate it from 0 to 1?....:(

Answer 13

Since you know the anti derivative, let's call it F, is F(x) = arctan e^x + C,

you simply need to evaluate F(1) - F(0)

but because you are subtracting the two functions, you don't have to work with the C, because C - C = 0

So it's arctan(e^1) - arctan(e^0) = arctan(e) - arctan(1)

Answer 14

what is arctan e^1???

Answer 15

haha, unfortunately my maths isn't that advanced yet :( but use your calculator to evaluate it.

Answer 16

okay, I think I may be able to leave the answer at that since their are no calculators on the quiz

Answer 17

oh, yes. Just leave it in this format:

\[\arctan (e) - \arctan (1)\]

Do you understand all the steps? I'll elaborate if there is anything that's poorly explained.