Trigonometric Integrals and Trig Substitution

Start   You should have finished the second web assign homework as well as the paper homework thats up.
its on the class schedule webpage.
thats where they will always be.
if youve had trouble getting into web assign please first try and contact them and then let me know and i will yell at them.
i know some people had trouble.
any other questions?
0:31also if youre having if when you look at your web assign uhh i think its problem 9 in homework 2 in particular has some issues with the grading.
i dont remember the answer.
im going to write something wrong i think its number 9.
i dont think its the sin but if the correct answer is something like i dont know, 1/2 sin^3(x)
1:02it takes an answer written this way and it does not take an answer written this way for some reason.
ive complained to web assign about it.
theyre going to fix it but its already out there.
so if you got burnt by this and lost points click on the ask your teacher link and get it fixed. i think its number 9 does anyone know?
the answer is really something like the cosine or cosecant or secant
1:30does anyone know?
yeah well so all of them should take this or this or this.
but one of them its either number 5 or number 9 i think its number 9 doesnt take this.
2:00and you put this in and it gets marked wrong you can change it to this and its right.
so if you put it in like this and it gets marked wrong and you change it to this and its marked right click on the ask your teacher and say web assign is dumb on this problem please give me my half a point back.
so you know the point is not to rob you for notation the point is to know you can do the problem.
so the only problem that i know of is that one.
2:31all the other ones will take any of those 3 formats.
okay so where we were is we were doing integration by parts.
which is useful when you have its a way of creating derivatives for integrals so if we have something
3:02so let me do one more example lets say the integral from-lets make it a definite integral this time no?
i feel like obama when the guy shouts out "you lie!" umm lets do something like this 0 to pi/4
3:35so suppose i want to do this integral.
theres actually several ways i could do this integral.
um one way i could do it so should i use parts on this?
yes? what should i take for parts?
take one part for sinx and the other part and take the other part for sinx?
4:05so that means that du is the cosine and v is -cos so that gives me minus did i do that right? yeah.
-sinx cosx from 0 to pi/4 minus the integral of v
4:33du so i think i did that right so far.
and this is from 0 to pi/4 and this is
5:03so sin of pi/4 is root 2 over..yeah?
well thats the other way you can do it by parts or you can do it by using the 1/2 angle identity.
either way is fine. so lets do it by parts though cause they wanted to do it by parts.
so the sin of pi/4 is root 2/2 cosine of pi/4 is root 2/2 so this is minus
5:32minus the sin of 0 is 0 the cos of 0 is 1 so that parts gone.
and now i have to do the integral of cos^2.
now if i do the integral of cos^2 i could do it by parts again.
but in fact its easier to remember a different identity.
actually i think if i do it by parts again i get stuck.
pretty sure i get back to the original integral if i do it by parts again i think.
6:03i get 0=0 or something like that.
so thats not going to work.
feel free to try it but im pretty sure it doesnt work.
so i have to use the fact to do this what is another way i could write cos^2?
right so remember that sin^2+cos^2=1 and so this becomes the integral from 0 to pi/4
6:30of 1-sin^2.
and now again we're in one of these tricks were in one of these tricks where im going to get the integral sort of related to itself so this is so this is uh 2/4 so this is 1/2 from here this is the integral from 0 to pi/4
7:00of dx minus the integral from 0 to pi/4 of sin^2 and that equals the integral from 0 to pi/4 of sin^2 so i have a thing that i dont know equals to a number plus something i can do easily minus the same thing.
so that means
7:32so that means i could bring this and add it to this and get 2 times the integral from 0 to pi/4 of the sin^2 = to 1/2 -1/2 + all the integrals from 0 to pi/4 dx is 0 is x
8:02from 0 ti pi/4 so that means so this is pi/4 -1/2 = twice the integral i want so the integral i want is pi/8 -1/4.
or pi-2/8
8:36which is a little bigger than 8 ok?
so theres 2 things going on here in this example.
1 im doing a integration by parts which is a definite integral and one of the things that you can do you dont have to you could go all the way to the end and then plug in is you could plug in as you go
9:01and get rid of some of the parts.
so thats what i get here.
the other thing is if i had tried to do this cos^2 the same way i did the sin^2 i would actually wind up with 0=0 so that doesnt work.
so sometimes when you do integration by parts you wind up with something true...0 does = 0 but you wind up with something that doesnt really get you anywhere.
9:33so this is true of integration in general going backwards is often harder than going forwards but its kind of a part of learning how to do these things.
it takes perseverance okay so im gonna stop with doing integration by parts for now cause im tired of it and we have to move along.
10:01cause we're still behind from the hurricane.
but as somebody else noted i could also do this integral by using well i want to do the easy ones first.
so suppose instead of that i had something like the integral of sin^3 of x dx.
and lets just make it an indefinite integral this time.
uhh lets make it easier let me throw a cos in there.
10:37say i have an integral like this.
what would i do?
so i hear people mumbling substitution.
factor the sin off?
11:07so why would i do that?
just cause i can?
i mean its true i dont know that it helps uhh so what these other people were shouting out is true lets make a substitution.
yeah i could do this but i dont want to.
but lets make a substitution what should i substitute?
11:32i hear a bunch of mumbles okay so i should substitute sin why should i substitute sin?
cause the derivative is sitting right here.
so i make the substitution just so you know call it u substitution.
im going to make the subsitution w equals sin of x.
and thats good because right there
12:01is my dw.
and so when i make that substitution that makes this a really easy integral.
and so this is just 1/4 w^4 and w was the sin so let me point out something that students always do
12:31because they think its quicker.
its really important that when you write these down that its true.
especially in something like this.
notice that everything i wrote equals the previous thing and i wrote equals every time.
and i didnt leave off bits until the end.
its important even though it makes me sound like im trying to be an evil accountant or something.
that you keep track of all of the bits because you need them later and so
13:01rather than just writing stuff all over the place i write it in order and make sure every stage is equal.
when you write = it means the same.
it doesnt mean becomes.
equals means equals means the same.
if something becomes something else dont write an equals sign.
also dont write arrow when you mean equal i see a lot of students that would write you know, this
13:30arrow this, arrow this.
in this case it doesnt matter whether you think of equal as a process that gives an outcome its just its group you just say this equals that and this equals that dont write equals if its not equal cause then you get in trouble.
you dont get in trouble from me you get in trouble because you wrote garbage.
and eventually you get angry.
sorry that was a little off to the side tirade.
i see it all the time in calculus.
14:00students just write junk all over the page and then magically the answer appears.
and thats fine if your junk has no mistakes.
but if you make a mistake we cant find it so we cant give you partial credit.
also youre less likely to make mistakes if you see it in an orderly fashion.
if youre going to write junk all over the page write it on another page.
okay i lost track of where i was going cause i had to rant for a minute, here we go.
14:32alright so this is actually a general trick.
when we see an integral with a pile of sins and 1 cosine laying around its easy.
and when we see an integral with a pile of cosines and 1 sine laying around, its easy.
so if we had an integral like
15:01sin^5 x cos^2 x dx lets make it sin^3.
suppose i had what am i doing?
suppose i have something like sin^3 x
15:31cos^2 x dx im going to make a substitution.
so im going to make this a clicker question.
16:24okay so choices are make some manipulations and then make the substitution u=sin
16:32make manipulations then make the substitution u=cos.
you can do either one and get the answer or no this is a bad idea.
so most of the people think we shoud manipulate it and then substitute for sine.
okay so lets see.
17:03what manipulations should i make?
those of you that think we should do a manipulation and then substitute for sin.
uhh since 57% of you think thats what i should do maybe at least 1 of you knows what i should do.
17:31so youre saying u=sin^3 du is then cos^2 x sin x dx.
wait a minute what am i doing? sorry.
im hallucinating..okay.
3sin^2 x
18:00well he said cubed.
so cubed isnt going to work.
because now i have an extra business.
so this is a bad idea.
u=sin^2 so then du is 2sin x cosx hey that it doesnt.
so if i do that
18:30so then that integral becomes well i have u then i have a sin ....x cos^2 x dx.
one of those is du so this is u sinx cosx du..i dont know what to do with that.
i guess this could be a square root of u
19:01but this guy is still there.
so this doesnt work so well either.
you in the back.
okay uhh.
so here im going to let im going to use the fact that cos^2 is 1-sin^2.
19:32and then?
okay i can do that and then?
yeah okay but i still cant do these integrals.
20:01and im still stuck, i dont know how to do either of those integrals.
okay lets just take this one soo... now i dont know.
im not trying to make fun of you.
im trying to point out there is something going on here
20:31and if i just tell you, youre going to forget.
so, anything else?
so this is a great idea except its a little bit wrong.
so right so peel off 2 of the sines and turn them into cosines.
this is a bad idea too.
21:00no thats a good idea.
so this is close and as he said, instead of focusing on trying to turn the cosine into sine lets turn most of the sines into cosines.
so write this integral as sin x and im just going to save it for later cause i might get hungry.
and then i have sin^2 x
21:30cos^2 x dx is the same but this is 1-cos^2.
so then all this becomes im going to write the sine at the end 1 or how about cos^2 x minus cos^4cx
22:02times the sinx dx.
and now im happy because when i eat these cosines i have the sine leftover for dessert.
i should let u be the cosine and so du is minus the sine that i need.
this becomes -du.
and this becomes u^2-u^4 which i can do which is easy.
22:47so now i get i lost my place.
the integral of u^2 minus u^4 du...whole thing negative.
23:01which is negative.. u^3/3-u^5/5 which is let me write it in the other order.
uhh u is cosine cos^5 minus cos^3 + a constant.
so that means that this choice for sure works.
23:37and i dont know how to make this one work.
and so if a doesnt work, neither does c.
so this is a general procedure.
whenever i have an odd power
24:00of a trig function laying around so in general if i have the integral of sin^m cos^n then if either one of these are odd if that is odd then i convert
24:33all but one sin into cosines.
by sin^2 is 1-cos^2 didnt leave enough room.
25:02if n is odd then i do the same thing but i convert sines into cosine.
by the other piece.
and if theyre both odd i can do it either way.
so this answer is sort of right in spirit but it depends on which power is odd and which one is even. yeah?
doesnt..dont care. be -2.
the trick is that i want to have laying around
25:33so what i want is the integral of stuff in sine..let me just call it a function of sine.
times the cosine x dx or.. some other function of cosines times sinx
i want everything except one of them to be the other guy.
and then this becomes easy.
straightforward substitution away we go.
so the thing is remember is not memorize this formula just remember that you know the most useful trig identity is not even trig its the pythagorean theorem.
this identity
26:36if you learn one thing from trig it should be that.
i hope you learn more than one thing from trig but thats the one thing that is the most important from trig.
which is just the pythagorean theorem.
so using that trick we can always turn you can always get rid of all but one of an odd power.
27:05we're running a little low on time let me not do another one of those.
you get to do some on the homework.
what case did i miss here?
did i stop this?
if theyre both even.
if theyre both even like the integral i did at the beginning
27:32or well lets just do that one.
so here the power of the cosine is 0.
thats an even number, the power of the sin is 2.
thats an even number.
i cant use this trick.
because if i take one sin away and if i turn both sines into cosines im in the same boat that i was in before.
and if i dont do it well then i have no hope.
28:01so i cant use that trick.
and i always forget which ones plus and which ones minus so i have to use a different trig identity so i want to use could do this identity or this identity maybe you remember and maybe you dont
28:32i always tend to forget them so i have to cheat, but i get to cheat.
and so here im going to use this one.
and then this integral becomes pretty easy.
so this is so i integration 1 and i get 1/2x
29:02and i integrate 1/2cos2x but when i integrate cos2x and i get 1/2sinx sin2x and then i get another constant so this is -1/4 check that you could check it by taking the derivative
29:31and then using this formula to go back.
did i mess up the sign somewhere?
k sooo if both are even then you use this then you convert so if you have an even number and
30:01notice what this does this takes an even power and it turns it into half of that power it changes the angle by a factor of 2 but it turns everything into half of that power.
and so ultimately if you take an even number and you keep dividing by 2 unless it was 0 eventually it becomes odd and then you can use this trick.
so if i wanted to do, say sin^4
30:31well then i would do it twice and get down to the power of 1.
so these things can be a little bit tedious.
but so i want to put aside this business for awhile these are easy once you get the hang of them other than sometimes theres a little bit of tedium with manipulating trig identities over and over and over again.
but i'll leave that for recitation.
31:09okay so another thing that we can do relate it to so what im trying to do right now is just go through techniques that work for various types of integrals.
so relate it to trig identities we have this one..sin^2+cos^2=1.
31:39so if i have an integral like watch im going to do one that doesnt work.
okay my brain just froze.
let me steal one from the book.
the problem with these is sometimes they dont
32:21so say i have one like this now here
32:31you try substitutions its not going to be so good well theres a right substitution you could make.
the trick to notice here is actually theres some trigonometry going on here.
just doesnt look like it.
somebody said something.
so if, so so the problem here is this stupid square root of x^2+4
33:02and i'd like to get rid of that and id like it to be like a square root that i can do.
so i remember maybe that should be a 1 to make it easier.
if i make it be a 1 will that be easier for now?
i can do it when its 4 too but lets do it when its 1.
so here i remember some trig identities so this guy
33:36i remember this guy which is not useful here and i remember this guy and this one is useful here.
tan^2+ 1 is sec^2 which you can just get by mushing this one around.
34:00and here i have something squared +1 becomes a square.
so that means if this x were only a tangent then this would become just the secant.
and then i could do this integral so lets just say well i wish the x were a tangent.
so im going to let x be the tangent some angle i dont know let me call it a theta.
34:33and so that means that dx is the sec^2 d theta ok fine.
and so here by doing this umm i can now rewrite the integral.
to get so dx becomes the sec^2
35:02x^2 becomes a tan^2 the square root becomes sec^2 and now i can do a little trig manipulation so lets reduce this the square root of sec^2 was sec so this is let me just rewrite it slowly
35:32so i have a sec^2 on the top i picked up another secant on the bottom and i have a tan^2 here and so that i can keep track of everything i mean this is really 1/cos^2, this cancels this and you have cos on the top and blah blah blah im just writing everything in terms of sines and cosines so that i know where i am.
so the sec^2 is 1/cos so im going to put that on the bottom.
and actually first im going to cancel this guy with that.
36:03that gives me a 1/cos thats from this bit.
and then im left with well tangent is sin/cos so that gives me a cos on the bottom and a sin on top.
squared so i think i did that right.
is everyone clear on where i went from here? yeah?
so this x^2+1 becomes the sec^2 im taking the square root of sec^2.
square root of something squared is just the thing.
and then this secant cancels one of those secants.
and then the secant on the top becomes a cos on the bottom umm this is upside down isnt it?
this is a cos^2 on the top
37:02and a sin^2 on the bottom so again i can cancel this with that.
and so and so after all that hoo ha foo faroo you wind up with
37:31i have the integral of cos/sin^2.
well thats one we can do, yeah?
its on the bottom.
so this is sin/cos but its on the bottom so i need cos/sin when i bring it up.
38:00but after all this mess i get that.
and this is easy.
what do i do?
u=sin. well, yeah i didnt use u yet and so this guy becomes just the integral of 1/u^2
which is uhh 1/u..negative but u well u is the sin
39:02so this is the cosecant..negative cosecant if you want.
but my question was not about theta my question was about x.
so if x is the tangent then theta is the arc tangent
39:32so this is really the csc of the arc tan but the csc of the arc tan has another name.
so we should figure out what that is.
why dont we draw a picture?
im going to draw a right triangle so you have to remember just trigonometry now.
and im going to let this angle be theta.
now if this angle is theta
40:02x is the tangent of theta cause that was the choice i made so tangent is the opposite over the adjacent.
and i want the tangent of this angle to be x so if this is x and this is 1 then the tangent of theta is x.
that means that this hypotenuse i use the pythagorean theorem is this squared+ this squared, take the square root.
40:39now what i want is not the tangent i want the cosecant, the one over the sin.
1/sin so the sin here, is this over this.
1/sin is this over this.
so this is minus
41:01x is minus square root of x^2 +1 over x.
which seems like magic but its reasonable. im integrating some algebraic function i do a bunch of stuff involved with trigonometry and eventually i get back to another algebraic function.
41:33so the short of this, well i dont know these are never short so what went on here is when we see square root or actually just something squared plus a constant this trig identity can be helpful and you turn it into a trigonometric integral which then is not so bad.
42:01if you do just the way i said how to do them.
and then you go back.
if i see something involving for example 1-x^2 i would use this identity.
and if i see something involving x^2-1 i wouldnt use the other one.
so lets look at this as x^2+4 so i wont go through the whole business
42:34and im going to do exactly the same integral but when its x^2+4 well im not actually going to do the integral.
im going to start it.
okay so this square + the thing tells me that i want to use
43:01whered you go guy in the glasses?
ok thanks.
it helps to look at somebody.
this tells me that i want to use something like arc tan but if i use this one its not going to help so what i want to do is divide everything by 4 and multiply back by 4.
so i rewrite this..which i dont have to rewrite explicitly but ill rewrite it in my mind or ill write it explicitly so you can see my mind.
43:31as..well im going to factor this 4 out and so when i factor 4 out here i get x^2 over 4 but i can actually write x^2/4 as (x/2)^2 like that.
and so this is and then i could pull this 4 all the way out
44:05and now i have a thing squared + 1.
now i can use this trig identity im going to let x/2 be the tan.
and then i just go from there.
so i want to get rid of this squarey thing.
so i get the same
44:30well it isnt quite the same as this one ill pick up some factor of 4.
but its almost the same.
k let me not finish this, yeah?
yeah since its 5 and i need the square root of 5 down.
you can take the square root of any number.
so if this had been instead of a 4 how about a 41 well then this would be a 41 and then here this would have to be 41/2.
45:02but i want to take the square root of 41 here.
so this would be square root of 41 and this would be square root of 41 and this would be the square root of 41.
its the same.
and now then youve got square root floating around and blechhhh but okay.
45:40well its the same.
i could have done 2tan theta or here square root of 41 tan theta=x its the same.
i dont i dont see maybe im misunderstanding what youre suggesting
46:032tan theta so what do you mean by well lets put it back before the 41.
2 is a lot easier to write.
okay so its a 4 again what are you proposing i do now instead of doing this mish mosh?
no no
46:30i have to convert it into a secant.
so the trick here that i want i mean it is you really have to divide by the square root.
i dont know how to get around it.
so so i mean if i let x be 2tan theta thats okay.
but thats exactly what i did here.
i said let x/2 be tan theta.
47:03so that i could okay alright.
umm soo all of these things when we use these kinds of rules
47:30well go from there. umm we'll start with partial fractions on friday