Trigonometric Substitution - Free Educational videos for Students in K-12 | Lumos Learning

Trigonometric Substitution - Free Educational videos for Students in k-12


Trigonometric Substitution - By The Organic Chemistry Tutor



Transcript
00:0-1 in this video , we're going to talk about how
00:02 to find the indefinite integral . Using trigger metric substitution
00:07 . Now there's three forms that you need to be
00:09 familiar with . The first one is the square root
00:13 of a squared minus X squared . The 2nd 1
00:18 you want to look for is a squared plus X
00:23 squared inside a square root function . And the last
00:26 one the square root of X squared minus a square
00:33 . Now for the 1st 1 you need to substitute
00:36 X with a sign data where a is a constant
00:42 . And the reason for that is because one minus
00:44 sine squared is close line squared . For the 2nd
00:47 1 , replace X with a tangent data and the
00:52 reason for that is one plus tan squared is second
00:55 squared And for the last one replace X with a
00:59 sick and data because one 2nd squared -1 is tangent
01:06 squared . Now these are the three forms that you
01:10 need to look out for when using trigger metric substitution
01:14 . So let's work on an example problem . Let's
01:17 say if we have the square root of four minus
01:20 X squared divided by X squared , how can we
01:25 integrate this function ? So notice that we have the
01:29 form square root a square minus X squared . So
01:33 we can clearly see that A squared Is four which
01:36 means that a . Is equal to the square root
01:39 of four or 2 . Therefore we need to replace
01:43 X with a signed data . In this case X
01:47 has to be to sign better . So D .
01:50 X . is going to be the derivative of two
01:52 signed data . So that's to co sign data .
01:56 T theater . And so now we're going to have
02:01 the integral of four minus . Let's replace acts with
02:06 to sign data . So this is gonna be too
02:11 signed peter squared and then divided by X squared .
02:15 Which is also to sign data squared . Now let's
02:20 replace the DX with two co signed data data .
02:26 We can get rid of this for now . So
02:37 now we've got to do some math . So let's
02:40 perform some algebra techniques to simplify this expression two squared
02:46 is four . So two sine squared is going to
02:49 be four sine squared theta . And on the bottom
02:52 we're also going to have four signs square data .
03:00 Now what do you think we need to do at
03:01 this point ? Mhm . At this point We need
03:06 to take out a four inside the square root so
03:10 we can get one minus sine square . So we're
03:12 gonna have the square root of four times one minus
03:16 sine squared theta . Yeah . Yeah . Now something
03:23 else that we can do Is that we can cancel
03:28 to to over four Reduces to 1/2 . So there's
03:31 gonna be two left over on the bottom and we
03:34 still have coast Sine theta . D . Theta .
03:38 Yeah . Now we can take the square root of
03:42 four . The square root of four . It's too
03:47 and then we can replace one minus sine squared with
03:50 co sine squared . So the coastline square part is
03:54 still inside the square root symbol . So now at
04:01 this point we can cancel 22 divided by two is
04:04 one and the square root of coastline square is cool
04:08 . Saint Boehner . So we have co sign and
04:11 this is supposed to be sine squared . Co sign
04:15 over sine squared times . Cosine theta . D .
04:18 Theta and co sign times co sign is Coulson square
04:32 . Now what do you think we should do at
04:33 this point ? The best thing I recommend doing at
04:38 this point is to replace coastline square away from one
04:41 minus sine square Because Science Square plus coastline squared is
04:45 one . Now at this point we can split the
04:49 fraction into two fractions . So we could divide one
04:55 by sine squared and we can divide sine squared by
05:00 itself . Yeah . Now you need to be familiar
05:06 with the reciprocal identities intrigue . one over sine is
05:09 Corsica , so one over sine squared is cosecha squared
05:15 and Science squared divided by sine squared is one .
05:18 So this is what we now have . Now .
05:20 What is the anti derivative of co Seacon square ?
05:23 The derivative of co tangent is a negative cosine squared
05:27 . So the anti derivative of negative cosine squared is
05:30 co tangent . So the anti derivative of positive Corsican
05:34 squared is negative co tangent And the anti derivative of
05:40 -1 d . Theta is going to be negative data
05:43 . And then plus scene . Now this is the
05:46 answer . It's the integral but not with the appropriate
05:49 variables because we started with an X . Variable and
05:53 now we have to change data back into an X
05:56 . Variable . So how can we do that ?
06:03 Now recall ? We said that X Is equal to
06:07 two signed data . So if we divide both sides
06:10 by tune , signed data Is X over two .
06:15 So we can make a right triangle . Now you
06:18 need to be familiar with the principles of Socotra ,
06:27 the soul part of sackatoga tells us that sine theta
06:30 is equal to the opposite side , divided by the
06:33 adjacent side . So let's place the angle theta here
06:37 , so opposite to theater is X . That's on
06:40 top divided by the ipod owners . So the hypothesis
06:43 two . Now we've got to find the missing side
06:47 . So whenever you have a right triangle , you
06:48 can use the pythagorean theorem . C squared is equal
06:50 to a squared plus B squared C . Is the
06:53 hypotenuse , which is to we could say is X
06:57 and B is the miss inside that we're looking for
07:00 . So two squared is four . And if we
07:02 subtract both sides by X squared , we're gonna have
07:04 four minus X squared is equal to B squared .
07:07 So the missing side B is going to be for
07:10 the square root of four minus X squared . So
07:13 we can put that here . Now if science data
07:24 is X divided by two , what is tangent data
07:27 based on circle ? Tour , tangent is opposite over
07:31 adjacent , this is opposite , this is adjacent .
07:34 So tangent will be X over the square root of
07:39 four minus X squared . Yeah co tangent is the
07:45 reciprocal of tangent . It's one over tangent . So
07:50 if tangent is X divided by the square root of
07:53 four minus x square . Co tangent , it's gonna
07:56 be the reciprocal of that fraction . So it's gonna
07:58 be the square of four minus X squared over X
08:02 . So now what about feta , what can we
08:04 replace theater with ? Now recall that sine theta is
08:13 X divided by two . So if we take the
08:15 arc sine of both sides , what's going to happen
08:19 ? What is the arc sine of signed data ?
08:25 Well these two expressions will cancel . And so we
08:28 can say that fada is the arc sine of X
08:33 over two . So we have negative data . So
08:35 it's gonna be negative arc sine X divided by two
08:39 and then plus C . So this is the final
08:42 answer . So that's how you can find the indefinite
08:47 integral . Using trig substitution . Now let's work on
08:52 finding the integral of X cube divided by the square
08:59 root of X squared plus nine . So we have
09:07 the form X squared plus a square . Or you
09:10 can write it as a square plus X squared five
09:14 plus three and three plus five is the same .
09:17 So what should we replace acts with If we see
09:21 this particular form in this case we need to use
09:24 the expression X . Is equal to a tangent data
09:29 . So a squared is the same as a nine
09:35 . So if a squared is equal to nine ,
09:37 that means A is equal to three . Which means
09:41 we should replace X with three tangent . Theta .
09:47 So let's go ahead and do that . Now let's
09:54 calculate detox . The derivative of tangent is second square
09:59 . So dx is going to be three seconds squared
10:01 theta D fader . So on top we can replace
10:05 XQ with three tangent beta Raised to the 3rd power
10:14 and then X squared . It's going to be three
10:18 tangent beta squared plus nine . And so d access
10:23 three seek and square data detailer . Now three to
10:30 the third power is 27 . So on top we
10:32 have 27 tangent cube . And on the bottom three
10:37 squared is nine . So we're gonna have the square
10:39 root of nine , tangent squared . Theta plus nine
10:44 . And then we still have three seconds squared fed
10:47 to the theater . Now in the denominator . Inside
10:58 the square root Let's take out a nine . So
11:01 we're gonna have the square root of nine . And
11:03 then after we factor out the G C . F
11:05 we're going to be left over with tangent squared plus
11:08 one and everything else . I'm just going to leave
11:12 it the way it is for now . Yeah .
11:26 Now if you recall one plus tangent squared is seeking
11:31 square And the square root of nine is 3 .
11:35 So we now have this expression . So now at
11:49 this point we can cancel three and a square root
11:52 of Seacon square Jessica . So this is what we
11:56 now have 27 tangent cube times . Seek and square
12:02 fattah D . Theater divided by seeking . So now
12:07 at this point we could cancel a second . And
12:15 so now we're left with the integral of 27 tangent
12:24 cube seeking to . So what can we do to
12:28 integrate this expression ? What we have here is a
12:33 trick biometric instagram . And instead of writing tangent cube
12:38 I'm going to replace it with tangent data times .
12:42 Actually tangent squared data times tangent data . Let's write
12:47 it like that . Now we need to perform another
12:51 substitution particularly use substitution at this point . So I'm
12:56 gonna make you equal to tangent beta . And the
13:01 reason why I want to do that is so that
13:02 D . U . Will be actually no that's not
13:06 gonna work . D . You will be seeking squared
13:10 . Mhm . I need to change it up a
13:14 bit . Let's replace tan squared With 2nd squared -1
13:21 . Because one plus tangent squared is seeking squared .
13:27 Now in this format we can replace you with sick
13:30 and theater so that D . U . The derivative
13:34 of C . Can and will be C . Can't
13:37 tangent . So it's gonna be seeking tangent theta .
13:41 D . Theta so I can replace second with you
13:50 . So let me get rid of this first .
13:55 And so now this is going to be 27 integral
13:58 of U squared minus one . And then tangent C
14:05 can't defeat to is the same as do you ?
14:10 So this becomes to you . You can see that
14:12 here these two expressions are exactly the same . Now
14:25 the anti derivative of U squared , That's going to
14:28 be used to the 3rd power divided by three .
14:31 And the anti derivative of one is simply you .
14:35 And keep in mind we said U . Is equal
14:36 to seek him . So we now have is 27
14:40 Times seeking to the third power divided by three minus
14:44 C . Can't data . And let's not forget plus
14:47 C . So we could simplify that 27 divided by
14:52 three is nine . So we have nine . Seek
14:54 out to the third power And then if we distribute
14:57 to 27 that's gonna be minus 27 times seek and
15:01 data plus C . Now the last thing we need
15:06 to do is convert that expression replace data with X
15:12 . Somehow . Now the first substitution that we made
15:16 was that acts is equal to three tangent beta Dividing
15:20 both sides by three . Acts over three is tangent
15:24 . So now we can make our right triangle .
15:29 So this is going to be fatal . And here's
15:32 the right angle . Now based on Socotra , tangent
15:35 is opposite over adjacent so opposite to theater is X
15:44 . And adjacent to it right next to his dream
15:47 . So now we gotta find a missing side .
15:48 So using the pythagorean theorem , C squared is a
15:51 squared plus B squared . We could say a stream
15:54 be his ex . So C squared is going to
15:58 be nine plus X squared . And to solve for
16:01 C we gotta take the square root of both sides
16:04 . So the third side of the triangle is the
16:07 square root of nine plus X squared . So now
16:11 we can evaluate second there using a triangle . But
16:15 let's evaluate co sign . The reciprocal of seeking .
16:19 Now based on sackatoga . Co signed data is adjacent
16:23 over hypotenuse adjacent history . So it's gonna be three
16:27 over the square root of nine plus X . Square
16:31 . So sick and theater , which is one divided
16:34 by co sign , is the reciprocal of this fraction
16:37 . That's gonna be the square root of nine plus
16:40 X squared over three . So now what I'm gonna
16:51 do is I'm going to take out 9 2nd Theatre
16:56 . I'm going to factor out that expression . So
16:59 it's nine seconds to times second squared minus stream plus
17:07 scene . Mhm . Now let's replay 2nd with the
17:14 square root of nine plus X . Square over three
17:19 . So seek and squared . We're gonna have to
17:23 square this expression And then that's going to be -1
17:28 plus E . So all we gotta do is simplify
17:32 what we now have . So nine divided by three
17:38 history . So we have three in front and then
17:40 square root nine plus X squared . Now , once
17:45 we square the square root of nine plus X squared
17:48 , that will simply be nine plus X squared .
17:51 On the bottom . We have three squared . So
17:53 that's gonna be nine . And then Ministry now let's
18:03 get common denominators . So negative three or negative 3/1
18:08 . I'm going to multiply the top and bottom by
18:10 three . And so inside the bracket I'm gonna have
18:17 nine plus X squared divided by nine . Actually need
18:21 to multiply top and bottom by nine and not three
18:25 . What was I thinking ? So it's going to
18:26 be -3 times nine , which is negative 27 And
18:31 then one times 9 . That's going to be nine
18:36 . I want to turn this into a single fraction
18:47 . Mhm . So I have nine plus X squared
18:54 minus 27 Divided by nine plus scene . Now 3/9
19:01 that reduces to one third so I can get rid
19:04 of the fraction inside the bracket . So I have
19:07 a one third outside and then square root nine plus
19:10 X squared and then nine minus 27 is negative 18
19:15 . So I have X squared minus 18 plus C
19:20 . And so this is the final answer for this
19:24 example .
Summarizer

DESCRIPTION:

This calculus video tutorial provides a basic introduction into trigonometric substitution. It explains when to substitute x with sin, cos, or sec. It also explains how to perform a change of variables using u-substitution integration techniques and how to use right triangle trigonometry with sohcahtoa to convert back from angles in the form of theta to an x variable. There's plenty of examples and practice problems in this lesson.

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Trigonometric Substitution is a free educational video by The Organic Chemistry Tutor.

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