Question

Evaluate the integral sec(x/7)dx
I'm always bad at trigonometric functions T_T

Answer 1

\[\Large \int\limits \sec\left(\frac{x}{7}\right)\;dx\]
Mmm this one is pretty tricky actually :U

Answer 2

It's hard if you don't quote know the secant integral yet.

Answer 3

We have to do this little sneaky multiplication (which is really not obvious).
You probably wouldn't know to do this unless you've done the problem before. :)
So don't worry.

We want to multiply the top and bottom by \(\Large \dfrac{\sec\left(\frac{x}{7}\right)+\tan\left(\frac{x}{7}\right)}{\sec\left(\frac{x}{7}\right)+\tan\left(\frac{x}{7}\right)}\)

Answer 4

Yeah, this is an integration that usually gets memorized straightforward.

Answer 5

Isn't there a way to do it by parts?

Answer 6

Gives us something like this yes? :U
\[\Large \int\limits \frac{\sec^2\left(\frac{x}{7}\right)+\sec\left(\frac{x}{7}\right)\tan\left(\frac{x}{7}\right)}{\sec\left(\frac{x}{7}\right)+\tan\left(\frac{x}{7}\right)}\;dx\]
From here it's a simple U-sub.

Answer 7

By parts? :u hmm i've never tried that before :3

Answer 8

Following along beau? +_+

Answer 9

yeah, I'm wondering if this is an only way?

Answer 10

It's the simplest way, of that I'm pretty certain.

Answer 11

As wio mentioned before, this is really one that you just want to memorize.
If you're being asked to go through the steps though, this is what we need to do. :)

Answer 12

Yeah, without memorizing the integration of it, this is the way you would go about it. For the most part, just try to memorize:
\[\int\limits_{}^{}\sec(u)=\ln|\sec(u) +\tan(u)| +C\]

Its just in this case your u is x/7. So if you solve for dx, you get du = (1/7)dx and dx = 7du. So this is just a basic secant integral that gets multiplied by 7.

Answer 13

The way I remember the formula is I know the derivative of secant is sec(x)tan(x). So just take that, throw it in an ln and put a plus sign in between.

Answer 14

ok thank you guys <3

Answer 15

Yep. And before you go, though, this is the same thing youd have to do if it were integral of csc instead. So just to remember that integration:
\[\int\limits_{}^{}\csc(u) = -\ln|\cot(u) + \csc(u)| + C\]
As for memorizing this one, its the same process. The derivative of csc is -csc(x)cot(x). So just keep the negative outside, throw the csc(x)cot(x) inside of the ln and throw a plus sign in between.

Answer 16

lol you're my saver <3
csc is the next homework i have to solve xD

Answer 17

Ah xD Lol, yikes. Have you learned how to integrate tangent and cotangent by chance?

Answer 18

yes i think i'm ok with it now ^_^

Answer 19

Alright, awesome then xD Thats all 6 of the basic trig functions then, so until you come across something funky youre all good then I hope. lol.