Question

Does the integration of sec^3 x start with the derivative of sec x tan x? I've heard that this integration problem is difficult, so I decided to try it out. Please just tell me if I'm on the right track; I don't need a walkthrough. THanks :D

Answer 1

Not sec x tan x, just sec x. :D

Answer 2

The derivative of sec x?

Answer 3

Yes. It's an integration by parts problem where u= sec(x).

Answer 4

but but but http://www.wolframalpha.com/input/?i=derive+sec+x+tan+x

Answer 5

I'ma be a scumbag and try it with the derivative of sec x tan x.

Answer 6

It's okay to do that if you know the antiderivative of sec(x).

Answer 7

Yes, I do.

Answer 8

Then it's a perfectly okay method. In fact, this is the first time I've seen that solution to finding the antiderivative of sec^3(x), so kudos to you! :D

Answer 9

Thanks, I came up with it (ironically enough) trying to find the antiderivative of sec x

Answer 10

(wanted to find the area between x=y and sqrt(x^2-1) and see if it was bounded as x approached infinity)

Answer 11

http://www.wolframalpha.com/input/?i=integrate+sec%5E3+x
Click on "Show steps"

Answer 12

I will NEVA click on that link! :D

Answer 13

So you were figuring out whether \[\int_{1}^{\infty} x-\sqrt{x^2-1}\ \text{d}x\] converged, right? So you used x=sec(theta) and lo, a wild sec^3(x) appeared. =))

Answer 14

No, I was doing the stupid, "Let's look for the derivatives of random functions and hope they come out to sec x!". Of course it didn't work (didn't even come close; I didn't use ln()), but derived sec x to get sec x tan x, then sec x tan x to get sec x(sec^2 x+tan^2 x), and realized that tan^2 x=sec^2 x-1, so I got sec x(2 sec^2 x-1), or sec^3 x-sec x

Answer 15

*2 sec^3 x- sec x

Answer 16

It's amazingly easy to use that to integrate sec^3 x; unfortunately, I did not know how to integrate sec x LOL

Answer 17

I eventually asked my sister, who had forgot, but later remembered to set u=(sec x+tan x)du

Answer 18

then, the usual integral du/u, or ln|u|, which resubstituted into ln|sec x+tan x|

Answer 19

@blockcolder

Answer 20

Yeah, and you'd get the formula for integral of sec^3(x). =))

Answer 21

I was amazed by the discovery actually; that a little bit of playing around gives you the answer to a "very hard" integration problem.

Answer 22

Anyone else got easy ways to do this integration method?

Answer 23

look in the back page of your book for the "trig reduction formula" and you will find it