Question

Techniques of integration problem, please explain? Picture of question and solution attached

Answer 1

My problem is I don't understand which properties they used to get
1/6(theta + sin(theta)*cos(theta))

Answer 2

Well if you differentiate that you get

\[1/6(1+ \cos^2 \theta - \sin^2 \theta)\]
and the pythagorean identity rearranged is: \[1-\sin^2 \theta = \cos^2 \theta\]

So you end up back at

\[1/6(2 \cos^2 \theta)\]

Which is the integral you wanted to solve. That make sense how we reverse engineered it? Hopefully this trick comes in handy for you checking your work in the future too, it's so easy and simple. :D

Answer 3

Could you break down the very first step you did even more? I think it would help me understand better. And also why do you say "if you differentiate", why would I being differentiating, and not substituting trig properties and integrating?

Answer 4

they skipped a raft of steps going from \[\int \cos^2(\theta)d\theta\] to \[\theta +\sin(\theta)\cos(\theta)\]
maybe you are just supposed to know that that is what you get when you integrate

Answer 5

I would really appreciate if you could explain the steps they skipped.

Answer 6

ok i can, but i bet $20 the integral is done in your book

Answer 7

i mean this one \[\int \cos^2(x)dx\]

Answer 8

it is one of those trig identities you though you would never use, or just skipped in pre- calc

Answer 9

I know that the cos^2(x) would become (1+cos(2x))/2, but then what?

Answer 10

oh ok if you know that you are done

Answer 11

\[\frac{1}{2}\int 1+\cos(2x)dx=\frac{1}{2}(x+\frac{1}{2}\sin(2x))\]

Answer 12

where do they get the sin(theta) in 1/6(theta + sin(theta)*cos(theta))

Answer 13

let me go back and look

Answer 14

oh that is another trig identity

Answer 15

\[\sin(2x)=2\cos(x)\sin(x)\]

Answer 16

the twos' cancel, get \[\frac{1}{2}\sin(2x)=\cos(x)\sin(x)\] they really make this more complicated than necessary by changing it

Answer 17

Do you know a website that has these identities listed? Because the websites I've seen are always missing something.

Answer 18

yeah of course because there are tons of them

Answer 19

i bet they are in your text, let me see how many are on the cheat sheet i use

Answer 20

got a bunch here

Answer 21

if you see them make a change like that (they already integrated, so nothing else has occurred) you should look for an appropiate identity that makes it work

Answer 22

can yall go view my question and help me @satellite73