Stony Brook MAT 126 Spring 2016
Lecture 11b: More trig integrals and trig substitutions
March 6, 2017

Start   okay but if you have sin^2(x)+cos^2(x)=1 that means sin^2(x) is 1-cos^2(x) and cos^2(x) is 1-sin^2(x).
you can manipulate that and get... uhh... i guess well do it this way.
tan^2(x) is sec^2(x)-1 and cot^2(x)
0:32i'll move out of the way in a second is csc^2(x)-1.
sometimes youll be substituting for sec^2(x) thats 1+tan^2(x) sometimes youll be substituting for csc^2(x) thats 1+cot^2(x) you should be able to move the one back and forth im not going to do it for you.
sometimes you will instead of using sin^2 = this and cos^2 = this you use a different substitution.
1:01the double angle formula substitution.
where cos^2(x) is 1/2 (1+cos(2x)) and sin^2(x) is 1/2(1-cos(2x)).
how do we know which one to use? we're going to go through that in a minute.
those are the 6 main trig substitution thingys youre going to want to be able to do.
1:31now how do we know which ones?
you basically have 3 categories of trig.
youre going to have sin(mx) cos(nx) where both are even both are odd and guess what the third one is going to be?
one of each.
when both are even youre going to use these.
now when i say both by the way if you only have a sine power or a cosine power its going to be the same kind of rules.
if both if its sine to an even power or cosine to an even power or both of these are even youre going to use this substitution.
so these are the even ones.
and these are the odd ones.
2:30and remember sin^2(x) is just 1-cos^2(x) and cos^2(x) is just 1-sin^2(x) soo.... when you have one in the integral you can always turn it into the other one.
K? its just a little flipping around.
im going to put these lights on.
i dont know if thats better or worse for the camera.
so lets do some of each flavor.
so lets say i have
3:01sin^5(x) cos^3(x)dx so notice theyre both odd.
so when both powers are odd youre going to end up using one of these trig substitutions.
now the truth is you could substitute for either one but since cos^3 is a lower power than sin^5 lets rewrite the integral and work on cos^3.
3:32so cosine so we leave the sin^5 the same and cos^3 now were going to want to use our cos^2 substitution so lets make it cos^2(x)cos(x).
why do we want to do this?
we want to do this because what weve done now is we got a cosine by itself.
the derivative of sin is cos, the derivative of cos is -sin.
so if everything else is sin
4:02then we could just do u-substitution cause these are all going to be sines and thatll be our du.
so rewrite that cos^2(x) as 1-sin^2(x) times cos(x)dx.
so again, how do i know to do this one?
because i see the odd powers.
and by the way it wouldnt matter by the way if sin was even
4:30that still would be fine okay cause im going to be rewriting this one.
thats where the odd power is going to help out.
alright lets do a u substitution.
i'll wait until everyones caught up.
let u=sinx and du=cosx. How do i know to let u be sin?
cause these will all be u's and that will be du.
5:01okay i dont want to have du to a power that wouldnt make any sense.
the general rule is the higher power is equal to u you certainly could have done it the other way its just messier.
cause remember, in the end sine and cosine are really the same thing.
you remember the sine curve right? sine curve does this and cosine curve does that its just shifted.
so theyre really kind of the same function i mean theyre not they kind of are friends so inside the integrand it doesnt really matter so its just easier if you pick the lower power.
5:35say it again?
that should be correct.
its 1/2(1+cos(2x)) 1/2(1-cos(2x)).
i hope its right.
alright so lets substitute in.
so sin^5(x) is u^5.
1-sin^2(x) is 1-u^2.
6:02and cosx dx is du.
dont forget the dx thing.
cause sometimes youre going to be substituting for dx like in integration by parts.
so everything has to get substituted.
this is a simple integral.
simple for me but it should generally be simple for you.
distribute that u^5 and we can integrate that, you can always do powers right? any polynomial.
6:31thats u^6 over 6 minus u^8/8 + a constant and then substitute back.
so its sin^6(x)/6 -sin^8(x)/8 plus a constant --- Shannon?
are we zoomed in enough?
so apparently the last video was just back a little too far.
7:09yeah if i wanted to do it-okay here i dont want to mess with your heads but i could have made it a sin^4(x) times sin(x) times cos^3(x) then id have sin^2 and sin^2 so what happened is i would end up with a 1-u^2 times another 1-u^2.
which is messier.
so thats why you want u to be the higher power
7:30and du to be the lower power.
just like they told you in church always go with the higher power.
so far so good?
some of you may feel you need that higher power after this course is over.
alright so we did an even odd one.
now what about odd odd well this is odd odd sorry if this was sin^6 it wouldnt have made any difference would it?
cause you would just have a u^6 here.
so it doesnt really matter
8:00if you have even odd or both odd.
the key is when one of them is odd.
so im going to have you practice one and then we're going to do an even even.
lets do cos^7(x)
8:32sin^3(x) dx.
so we'll practice for a couple minutes.
make sure you've got this down and im going to leave the rest here in case you need to have a reference.
we ready?
this is just like the last one all you have to do is rewrite this.
so if you see the sin^3
9:00this is now sin^2 and sine.
and again i could have changed the cos^7 but thats much more work.
and now im going to do almost the same substitution except its going to be the other way.
so u is cosx du is -sinx.
the negative isnt really that big a deal but gotta get it right.
so we now have
9:40cos^7(x) is going to be u^7 oh i didnt make the switch yet sorry i left out a step.
cos^7(x) sin^2 will be 1-cos^2 sinx dx.
now we can substitute in everything.
so the 1-cos^2 is 1-u^2
10:02and sin x becomes -du.
you get the minus signs?
yeah you hated that.
so this is -u^7 +u^9 du.
thats -u^8/8
10:31+u^10/10 + the constant and then put the cosine back for u.
so you should be feeling better about these now.
k we like these now?
ehh its okay its not great.
how we doin so far?
you know whats coming next? evens.
11:01evens are a lot less fun.
the even rule applies to whether you have just a cosine or a sine or you have both cosine and sine.
and these are messier so we generally wont give you-we cant give you a very high power.
its too much work.
but we can give you something like sin^4(x) dx.
11:34see now heres the thing.
suppose you took out a sinx like we did on the last one.
youd be left with a sin^3.
you dont have a substitution for sin^3.
you could have a sin^2 but you'd be left with a sine somewhere in there.
so you cant use this substitution.
so you have to use this substitution which is a lot less fun.
so this is sin^2(x)
and thats going to-now were going to take sin^2 and were going to replace it with 1-cos(2x) dx.
so we get 1/2(1-cos2x)
whose calling me? huh.
we all see that? you dont need the square brackets but theyre fancy.
im sorry?
question.. no question? ok.
people just sort of yell things out or maybe im losing my mind.
13:02so (1/2)^2 is 1/4 now you have to foil that out.
thats 1-2cos2x+cos^2(x) dx.
so remember what i told you about constants the quarter isnt important.
its just a quarter times the whole thing so you could put the quarter outside the integral.
13:31and we look at this integral and we say well the integral of 1 is easy thats just x.
the integral of cos(2x) is easy thats just going to be sin(2x)/2.
so this is the messy one.
because its cos^2(x) so lets break this into 3 integrals.
lets do 1/4 the integral of 1 dx.
-1/4(2)-1/2 the integral of cos2x dx.
14:01+1/4 the integral of cos^2(2x) dx.
now theres always somebodys whose a little behind who had their head down so the 1/2 is because its 1/4 times 2.
and we got that.
so i distributed the 1/4 and i broke this into 3 integrals 1/4(1) 1/4(2) and 1/4(1) and i have the 1, the 2cos2x and cos^2.... 2x.
are you with me so far?
14:37so this integral is just 1/4x and this integral is 1/2 sin(2x)/2 so thats going to end up becoming sin(2x)/4.
and now what do we do with this last integral?
well you go back and you look at the double angle formula here.
this is x this is 2x remember x is just a whatever.
so if this is 2x this is 2(2x) or 4x.
if this is 10x this will be 20x.
so how many x's you got here its just double here cause this comes from the double angle formula.
so it doesnt really matter cause a lot of people get lost on this.
thats x thats 2x so since here i have 2x im going to have 1/2
15:311+cos(4x) dx ...ok?
cause this angle is always double whatever that angle was.
so whatever the angle is we're doing the cos^2 its double when you dont do cos^2.
so notice what weve done with this integral ok? we had 4th power we broke that down into no power linear quadratic and now we take the quadratic and again we break it down into no power and linear.
16:02so now everything is going to be straightforward to do.
its easy for me to say.
its not everything is multiplied by 2.
what matt said, its just the angle thats multiplied by 2.
so basically however many this is this is doubling.
16:32alright so lets pull that 1/4 out so thats going to be 1/8.
the integral of 1 is x.
the integral of cos4x is sin4x/4.
so we've got 1/4x -sin(2x)/4
17:04+ that.
+c dont forget the +c.
how do we feel about this one? yes?
why do i have the 1/2 here?
that comes from the substitution again.
and the substitution is the double- the angle squared is a 1/2 of 1 plus cosine of double the angle.
17:30so here i have cos^2 that becomes 1/2(1+cos) and instead of squared of 2x it becomes cos4x.
so its tricky you have to keep track of both the minus signs and whats going on with the fractions.
okay? thats what makes these hard theyre just messy.
these arent fun, alright.
we'll practice another one of these.
and by the way if i make this sin^6 this would be (sin^2)^3
18:01it starts to get really annoying.
i dont want it to get too bad.
we wouldnt give you that you'd have to cube it with sin^10 youd have (sin^2)^5 and then there will be this mess to the 5th so youd have a long chain of them. there are actually simplification formulas for doing these so if you were doing them for real first of all you just put them in a computer and let wolfram alpha figure it out but wolfram alpha uses the formula.
18:31okay if youre bored you could sit down and you can just use uh you know various letters and youll see where the formula appears but most of you probably have better things to do.
maybe not all of you though.
alright so lets give you a similar one.
lets see how we do on that.
19:00so sin^2(4x) and cos^2(4x) dx.
give you guys the first step and then you can practice from there.
so sin^2(4x) is going to become 1/2 (1-cos8x) and cos^2(4x) is going to become 1/2 (1+cos8x)
so thats the first step.
and im going to let you work on it for a couple minutes.
lets multiply this out.
by the way some of you might recognize that this is essentially the double angle formula for sine.
cause sin2x is 2sinxcosx so you could play with this and sort of skip a step but its okay if you dont see that.
20:00so this becomes 1/2 x 1/2 is 1/4.
because you guys are all my little math geniuses here you say 1-cos^2 is sin^2(8x) dx.
now for sin^2(8x) im going to use this formula im going to get 1/2 (1-cos16x).
so this is going to be 1/4 1/2 1-cos 16x dx.
21:00thats equal to 1/8 okay.
integral of 1 is x and the integral of cos16x sin16x over 16 + a constant you can distribute the 1/8 or not i dont really care.
howd we do on this one?
not as intimidated as we were 45 minutes ago?
now lets move on to... im sorry?
21:34theres other ways to do this theres about 3 different attacks that will all work the same on this problem which you end up in the same spot.
okay yes?
how i got this?
thats 1-cos^2
arent these fun?
no one seems to think theyre fun.
thats so sad.
how are you ever going to be mathematicians if you dont think this is fun?
i know the answer to that.
lets move on from sines and cosines to secants and tangents.
22:36wait until we do the trig substitutions which we're doing on wednesday.
well we might get them in today might get a little in.
suppose i had something like that.
i believe we did this exact problem but we'll do it agian.
23:02i see tan^3sec^3 x now if i see sec^2 so dont rewrite this im just going to erase this for a second if this was sec^2 then this would be just a straight u substitution cause id let u be tan and du would be sec^2 but i have sec^3.
so what am i going to do?
cause i kind of want to have the sec^2 but
23:31i just have a secant leftover which doesnt really help.
however the derivative of secant is sectan so if i made this tan^2 x sec^2 x times secxtanx how do i know to do this?
now you know.
okay if you didnt know before now you do.
24:00now here the derivative of secant is sectan and remember for tan^2 i can replace it with sec^2-1.
so i can make this sec^2 x-1 that replaces the tan^2.
times sec^2 x times secxtanx
so again first i pulled out a secant and a tangent and then i have even powers so if i have even powers of secant and tangent i can take the tan^2's and replace them with the sec^2's so you start to see this depends a lot on when you have evens and odds.
so if i let u be secant du is secx
25:01tanx dx.
so this becomes the integral of u^2-1 times u^2 du.
distribute the u^2 and you get
25:30u^4 -u^2 du thats nice and simple.
remember the whole secxtanxdx gets replaced with du.
thats u^5/5 -u^3/3 so thats sec^5 x/5
26:01-sec^3 x/3 ok?
so far so good?
so how do i know to do that?
well as i said i dont see sec^2 i see sec^3 and that tells me that i can take out a secant and a tangent.
26:31like i said i gave you guys about 20 practice ones on that.
exactly 20 practice ones in that document.
so theres a lot of chance to do these.
later when other people are out having fun you can sit and do these.
make sure you call your mother and tell her how hard youre working.
make her feel bad.
what if its not an odd power lets try a different thing.
what if this was sec^4?
27:03any ideas?
actually wait, is that hard? lets see.
so what am i going to do?
thats right.
im going to do sec^2 sec^2.
why am i doing that?
because the derivative of tangent is sec^2 and i can take everything else in the integral and put it in terms of tangent.
27:35slowly but surely you should start to be seeing what we're trying to do each time.
this is tan^3 x sec^2 x oh yeah its supposed to be an x.
its alright.
thats the whole you know its there.
dont email me about that stuff.
28:04and contrary to popular belief i dont check my email at 2 in the morning.
so when youre emailing me at 2 in the morning im not going to respond.
if i had your phone number i would call you at about 6:30 in the morning to punish you.
but i dont...alright.
so as i said, this-one of these sec^2's will be the derivative of tangent so i could just take the other sec^2 and turn it into a tan.
right? because if tan^2 x is sec^2 x -1
28:32then sec^2 x is 1+tan^2 x.
so i can now make this tan^3 x 1+ tan^2 x times sec^2 x dx.
29:03why wouldnt i break up both sec^2's? i want to keep 1 for a derivative.
because i need i need to keep sec^2 x dx this is going to become du.
so now if i let u be tanx du is sec^2 x dx.
29:33so this now becomes u^3 (1+u^2) du.
why didnt i do u^3 and du^2?
because i dont know what to do with a du^2.
i only have to have out du.
30:01so to get everything in the integral in terms of something and just the one thing left would be du.
alright distribute and this is very straightforward from here.
thats u^3 + u^5 which is u^4/4 +u^6/6 and thats tan^4 x over 4 + tan^6 x/6
30:32+ a constant.
so again the controlling thing here is the power of secant.
and sec^2 sec^3 if its just secant we memorized what the integral of secx is remember?
we wrote those down ill put them on the board again.
the integral of tanx and the integral of secx the integral of tanx is negative
31:01ln cosx + a constant.
and the integral of secx is ln secx + tanx.
plus a constant, whatever im not going to put the constant.
you should already have those written down from the other day. yes?
and you should memorize the integral of ln x.
yeah ill put up a page of integrals you should know.
31:33spring vacation while the rest of you are off getting tan and not having alcohol cause youre under 21 i will be sitting in my office alone, cold and in the dark thinking of things to make your math life better.
eating cold beans from a can with no one there to love me but ill be okay.
but anyway so yes ill make you a page i already have it somewhere i just have to find it.
alright lets do something slightly different
32:02but related.
these really hurt your brain so brace yourselves.
im just going to erase everything you didnt get it down, watch the video.
copy from your friend.
32:31what if i wanted to do the integral of the square root 1+9x^2dx.
you look at this and you reach into your bag of tricks and you say u substitution. i could let u=1+9x^2.
the problem is du is 18x.
i dont have another x term.
what about that whole u and then the u equals something and then subtract nah thats not going to work either.
33:00theres no trigonometry in there.
integration by parts well if i let this be u thats really messy.
and that would be dv that just comes out x..i dont...what do i do?
heres what you do.
you make a substitution that does the following let tan theta = 3x.
how would i ever know to do that?
now you do okay?
33:32there are families of these i know this kept mathematicians busy for a couple hundred years.
each time somebody would come up and i got a new one for you and theyd wrack their brains for awhile and then somebody clever would figure it out and go ah! and theyd all say its easy after that.
so if i let this be tan theta now why? youll see a in a second then sec^2 theta d theta is 3 dx
34:02or 1/3sec^2 theta d theta is dx.
okay i picked a hard one to start.
my apologies.
now why would i do that substitution? well now if 3x is tan theta then tan^2 theta is 9x^2.
and that would become the square root of 1+ tan^2.
34:33and then for dx i put 1/3 sec^2 theta d theta.
and well spend more time on these on wednesday.
what is 1+tan^2?
its sec^2.
so this is now the square root of sec^2
35:01the 1/3 goes out here sec^2 theta d theta.
brace yourselves its going to take the whole board.
what is the square root of sec^2?
times sec^2 so this is now going to become 1/3 the integral of sec^3 theta d theta.
35:31alright now i have to integrate sec^3.
i did this at the end of the last class but we'll do it again.
theres a lot of people at that point their little heads hang first and then say im done they wrote it down but they werent really paying attention.
so the integral of sec^2 is the following im not going to worry about the 1/3 for the moment.
integral of sec^3 break that into sec theta
36:02sec^2 theta d theta.
now we do integration by parts.
why cant i do u substitution?
why cant i do any substitution?
well if i make this 1+tan^2 the problem is the derivative of tan is sec^2.
soo why dont i do this?
36:31let u equal secant dv sec^2 theta d theta.
du sec theta tan theta d theta.
v is tan theta.
so i let u be secant derivative of secant is sectan
37:03dv is sec^2 antiderivative of sec^2 is tan.
so now i have integral of sec^3 theta d theta is u times v which is sec theta tan theta minus the integral of sec theta
37:30tan^2 theta d theta.
and whats tan^2?
sec^2 theta -1.
so i get the integral of sec^3 theta d theta sec theta tan theta minus integral of sec theta
38:01times sec^2 theta minus 1 ive run out of space im going to have to squeeze this a bit.
well i could squeeze in 1 more row.
this now distribute you get sec theta tan theta minus the integral of sec^3 theta + the integral sec theta
38:30d theta.
and notice ive got the -sec^3 here and the sec^3 there.
so i can add to the other side and ill get 2 integral of sec^3 theta d theta= sec theta tan theta + integral of sec theta which is ln of sec theta + tan theta
39:00+ constant.
the whole thing is multiplied by a third so this becomes 1/3 gotta divide that by 2 1/2 im not even close to done. sec theta tan theta actually am close to done i know im just torturing you guys.
plus ln secant theta + tan theta
39:30+ c. so but wait okay so thats the integral.
i mean its big cause i wrote all over the board so lets start over and make sure everybody understands.
we had 1+tan^2 (1/3sec^2).
1+ tan^2 is sec^2 square root of sec^2 is secant so this became sec^3.
sorry i picked the hard one to start with.
the integral of sec^3 is going to be sec times sec^2
40:02do integration by parts let u=secant.
dv=sec^2 du=sectan v=tan.
so you can rewrite sec^3.
as sectan-sectan^2.
then you let tan^2 be sec^2-1.
distribute add the sec^3 to the other side and now youre just left with this one integral.
so we're all the way down to here we've done the integration
40:31and then theres just one fun thing you have to remember.
this isnt in terms of theta this integral is in terms of x.
so what do we do? cause we've done the whole thing but we've done the whole thing in terms of theta.
tan of theta is 3x.
and that means...draw the triangle.
the tan of theta you could say this is 3x.
41:01and thats 1. (SOH CAH TOA) thats the square root of 1+9x^2 which by the way is this.
so wherever you have secant theta you could substitute square root of 9...1+9x^2 for tan theta you could put in 3x square root of 1+ square root blah blah blah.
remember what i told you about derivatives and integrals integrals are much harder.
41:31you think this is bad. I didnt become a teacher to help you with this.
they just sort of assigned it and said good luck.
you get 1/6 sec theta which is square root of 1+9x^2 hypotenuse over adjacent.
times tan theta which is 3x plus the ln of the absolute value of sec theta
42:02+3x + a constant.
and i said you would never look at that and say oh well its just this..ok?
so this is why integration is so much more difficult than than differentiation ok?
cause the derivative of this isnt hard at all.
but the integral is ridiculous.
so i accidentally picked a really hard version so now im going to give you an easier one
42:30that we're going to do.
i know, i know.
look at the bright side a week from now you'll be in florida paradise island massapequa one of the garden spots of long island.
maybe if youre really lucky youll be in wyandanch.
its okay ive been to wyandanch.
ive been there twice to wyandanch its okay.
alright we ready?
43:02im sorry matt?
where do i plug back in? well once i know that i took tan theta and i said that was 3x right?
that means i have a triangle where the tangent of the triangle is 3x/1.
so i can find the secant and the sin and cosine and all of that.
lets do another one.
i can erase? good.
44:01arent you glad you came today?
your friends who didnt bother to come to class today you can make fun and them and say no i will not give you the notes.
you earned the notes today.
alright the square root of 1-4x^2.
im going to let sin theta =2x.
on wednesday ill show you why i picked these substitutions.
because this would now be 1-sin^2 okay then cos theta
44:35d theta is 2dx or 1/2cos theta d theta =dx.
so i substitute.
if sin theta is 2x
45:01then sin^2 theta is 4x^2.
so this becomes square root of 1-sin^2.
and dx is 1/2 cos theta d theta.
so far so good?
dont worry we'll show you the substitutions.
okay as i said theres pattern to this.
so all i did was i took 2x and i make it sin theta
45:31notice thats the square root of 4x^2 okay?
so square root of that i made sin theta so when i square it youre going to get that okay?
so whats 1-sin^2?
the same as cos^2 and the square root of cos^2 is cosine times cosine is cos^2 theta d theta.
and we know how to do cos^2 cause we did a bunch of those.
46:03so you do the substitution this is going to be 1/2 integral of 1/2 1+... cosine 2 theta d theta.
see why we learn the trig ones first? yes?
46:31remember this is the square root of cos^2.
okay? so 1-sin^2 is cos^2 square root of cos^2 is cosine.
times cosine.
again somebody thought this and everyone else was like wow i hadnt thought of that.
so this is a relatively easy integral for us.
the 1/2 times 1/2 gives us 1/4 you get theta + sin 2 theta/2
47:04+ a constant.
and now for the fun part.
remember this is in terms of theta.
so you have to go back and figure out sin and cos.
you say okay.
so i have a triangle sin of theta is 2x/1.
so the missing side the adjacent side is the square root of 1-4x^2.
it will come out to be this
47:31if you do it correctly.
so you get 1/4 so whats theta? theta is inverse sin of 2x or arc sin of 2x.
you wouldnt have seen that coming.
anddd.. sin of 2theta..ugh sin2theta sin 2 theta what do i do with sin 2 theta?
48:00heres another one you probably forgot.
lets put this over on our board here.
we'll squeeze in sin2x is 2sinxcosx.
you guys are supposed to remember this remember when you memorized them and you forgot them and then you memorized them and you forgot them?
one more memorization and youre probably good.
i know but the reagents gave us a formula sheet so why do we have to memorize them?
48:32i know its not fair.
i understand i have a head start on you guys.
so sin of 2 theta is 2sin theta cos theta so this is 2 sin theta cos theta over 2.
so its 1/4 arc sin 2x
49:01sin of theta is 2x cos theta is the square root of 1-4x^2 +c.
so how do you know which substitution to do?
well theres basically 3 sets.
we only did 2 of them today.
if you have a^2
49:31minus x^2 right?
youre going to let x= right youre going to let x/a=sin theta.
and im going to do these again on wednesday so you dont really need to write them down.
if you have a^2 + x^2 youre going to let x/a be tan theta
50:00and if you have x^2 minus a^2 x/a will be = to sec of theta.
i have to double check these okay?
but im 98% sure its correct but not 100.
but these work out very nicely.
so well practice some more of these on wednesday.
practice problems will be available at 5.