Question

integral of sec^3(theta)

Answer 1

any hint? Im guessing u have 2 multiply top/bottom by something

Answer 2

http://www.wolframalpha.com/input/?i=integral+of+secx%5E3dx
its a special reduction formula....

Answer 3

right.. but how u go about doing it ?

Answer 4

you don't need a reduction formula for this one, but the other way is a bit annoying

Answer 5

got, reasoned it out. thanks

Answer 6

\[\sec^3\theta=\sec^2\theta\cdot\sec\theta=(\tan^2+1)\sec\theta\]ok, if you got it already then congratulations

Answer 7

ye, i did it that way :P ty

Answer 8

or you could use integration by parts:
http://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx
example 10^

Answer 9

i meant inte. by part.

Answer 10

that is the better way, yeah
I'm just tired...

Answer 11

haahahaha ya, i kno the feeling

Answer 12

i got a curious question. For polar equations, When they ask u to find the point where it's horiz/verti tangents, y do they only wants the first few thetas?

Answer 13

by "the first few theta's" you mean in the interval [0,2pi] ?

Answer 14

no, they dont give u interval, just r=something, and that's it,
when i solved it, for ex. cos(theta)=1, so theta has many value, but they always pick the first 2-3 theta (and positive)

Answer 15

so i kept having extra answer, which im not sure even if it's right or wat.

Answer 16

they are choosing what are called the "principle values" of theta, which are the values that satisfy in the interval \([0,2\pi]\)
you can always add \(2\pi k| k\in\mathbb N\) but that goes for all values of theta in ases like this, so we ignore that part often and just write the "principle values"
i.e. those between [0,2pi]

Answer 17

in cases*

Answer 18

so those extra one i have r ok? or should i remove them?

Answer 19

wait.. after 2pi, it reapeats.. is that y ? u ignore the rest since it gona be same asnwer with different theta value?

Answer 20

for the most part, yes
if you had \(\frac\theta2\) as an argument it may be sightly different
depends on what they are...
this integral is indefinite, so it doesn't matter
but \(\cos(\frac\pi2-\theta)=\sin\theta\)
so you can use that identity as well
in short, there are infinitely many answers, because we can go around the unit circle infinitely many times and get the same value for \(\cos\theta\), so we often just write the "principle values" i.e. those in the interval \([0,2\pi]\)

Answer 21

\[\cos(\theta)=\cos(\theta+2\pi k);k\in\mathbb N\]but we exclude the second part often, because all it says is "go all the way around the unit circle and the trig function is the same" which is obvious

Answer 22

got cha, I still need to memorize the unit circle, which maybe a lot of help. well ty alot.

Answer 23

no prob :)