Up next in 10
How to Integrate Exponential Functions Using The Substitution method
16 views
Feb 19, 2025
Integrating exponential functions using the substitution method
View Video Transcript
0:01
subscribe to hey so we're going to let U
0:04
equal what let you equal x right yeah so
0:08
what's the derivative of Ro X without
0:13
thinking one
0:16
what 12 what are you saying X DX it's 12
0:22
right of x d if you do you can right now
0:26
we know that now we can split this right
0:29
so so we know that the uh
0:32
du 12 two I
0:35
mean 2 du will be equal to what 1 x DX
0:43
correct because I'm trying to get 1 s of
0:46
X DX by itself wait couldn't do you also
0:48
equal DX over two yeah yeah you can do
0:52
that too right and now we just go back
0:53
and just substitute that in here so that
0:55
gives us now what two outside and then e
0:58
to what E3 U d right because 1 / X DX is
1:03
2D and then of X is U it's just e to the
1:07
U
1:09
now are we good all right so the
1:13
anti-derivative will be what two e to
1:16
the U plus C and then we replace U by by
1:19
the of X so that we get 2 Eun x + C all
1:25
right now let's try and work on the next
1:27
one it's a little more complex what
1:30
again did you do the the next one yet
1:32
three no all right let's try that so we
1:36
got e to thex over 1 + what e tox DX
1:42
right what would be the easiest way to
1:44
solve this
1:47
problem what would you do let
1:52
= you should go let you equals the whole
1:55
thing because if you do this here it'll
1:56
be harder for you to like find the DU
1:58
right letal
2:00
e because the of one is what zero right
2:04
so let U we going to let u = 1 + e x e x
2:08
d x e to X I mean and what's
2:12
du what's the dtive of 1 0 what's the do
2:16
with e to the
2:19
x 1 orative e to the X DX right and now
2:24
we just go back and substitute that in
2:26
here so we have A1 so we get Negative
2:30
uh
2:31
duu over U right now what's the an of
2:35
one U du we did it what is
2:39
it y forgot what's one over U one
2:44
no natural log of absolute value natural
2:47
log of absolute value got negative
2:51
natural log of What U plus C right and
2:54
we know what U is U is what 1 + e x so
2:57
this is negative natural log of 1 + e x
3:02
+ C all right that make
3:08
sense yeah now the next one will be a
3:10
little more complicated cuz now we have
3:12
like different things there so let's try
3:14
and work on that too let me raise this
3:20
here so we have um 5 - e to
3:28
X over
3:31
e 2x DX
3:34
right so what do you
3:36
think the easiest thing to do here is I
3:40
think you should make U let = 5us no no
3:44
no let U equal e to the power right we
3:47
can do that
3:50
2 and remember we can split these two we
3:53
can split them if you want right because
3:56
if you don't split them it be hard for
3:58
us to solve it so it's better to split
3:59
them to begin with right so I'll start
4:01
by saying this is like 5 e 2x right
4:06
minus e x e 2x right DX now we know we
4:10
can take this what up can we yeah yeah
4:14
that that'll be um e to all right so
4:19
this gives us what 5 e to the what -2X
4:21
to right
4:23
minus E to
4:26
thex DX and now we can use two U
4:30
substitution to solve this here right we
4:33
can use two U substitution to solve this
4:36
now we can go let U here first let U
4:40
equals e
4:42
-2X right are we good oh you just cancel
4:46
out top of this yeah I just took it up
4:47
and then now becomes e x and let = e -2X
4:52
du gives you -2 right e to -2X DX right
5:00
and
5:01
then oh no no no I'm I'm doing too much
5:04
here hold on that was a mistake I
5:06
apologize so let U equal let U = -2X not
5:11
E2 what and then
5:14
D2 DX right so now we can do and then
5:18
here we're going to do the same thing
5:19
we're going to let u = x and then du
5:24
here will be1 DX and now we can just go
5:26
ahead and substitute that in there right
5:29
so now here we going to have this is
5:32
five outside we 5 over 5 2 e to U du U
5:39
and then minus here since you have a
5:42
negative and a negative becomes positive
5:44
or we just put ne1 here to make it easy
5:47
e to the U du and now we can find the
5:50
antiderivative of that right that'll be5
5:54
/ 2 e to the
5:58
U minus
6:00
or plus because we have two negative
6:01
plus e to U plus C and then substitute U
6:06
by -2X here and then sub U byx on this
6:10
one we don't have to put two C's just
6:13
one C is enough so that give this wait
6:16
why you put the U on the outside of the
6:19
which one on the first five five five
6:22
over wait shouldn't it be why I put the
6:24
the five on the outside yeah why did you
6:26
put the outside well cuz I I don't want
6:28
to I mean you can do that too you can
6:29
keep it in here it's not going to change
6:30
anything you can put it outside or
6:32
inside shouldn't the the in the
6:33
parentheses right there be -2 inste 1
6:36
here yeah no because it's 1 DX yeah but
6:40
it should be2 though right no that = X
6:42
for the second one I separated this into
6:44
two right for the for the left one we
6:47
let U = -2X and for the second one we
6:50
let u = x so we have two separate we
6:54
doing the same thing but twice right so
6:57
now this is e to -2X X Plus e
7:02
tox and then plus C and I should take
7:05
care of that all right are we
7:08
good all
7:14
right I also want to try one of them
7:16
with the co
7:18
cosine and then for the cosine let's not
7:21
worry about the corner point for now
7:24
okay are you good yeah we can't see bro
7:26
all right let me
7:28
move
7:34
I feels like so prop right now you feel
7:36
like what it's a proper like this that's
7:40
how you're supposed to see
7:41
anyway no yes you are yes you are it
7:45
good for your back you have to be
7:47
propped up like that because then you
7:49
don't Str your
7:53
leg all right y trying to work with this
7:56
one here so we have P sin pix
8:00
cosine PX DX you just make up a problem
8:03
no I didn't make it up it's on your
8:05
no
8:09
I she has one yeah yeah but for us let's
8:13
not worry about those let's just do like
8:15
this I I'll just put this on because I'm
8:17
not really concerned because for those
8:19
you just have to plug it in I'm just
8:21
worried about really understanding what
8:22
to do here first right so let's try and
8:25
find
8:27
this so why would you let U equal
8:31
here um you let U equal sin pi x good
8:36
let U equal sin Pi that's tough and
8:40
what's
8:46
du first PI right Pi cosine p x DX don't
8:51
forget the pi because you have the
8:54
inside right and now cosine XX DX is
8:57
what du over
9:00
P
9:02
right don't forget that okay don't
9:05
forget to take the pie out to divide it
9:08
by pi U so now you going to have 1 over
9:12
Pi e to the U du and then the rest is
9:17
just a piece of cake right you just find
9:19
the in
9:22
of
9:25
tough and then replace You by Co sin pix
9:28
so one over pi
9:32
i c all
9:37
right e to the pi x and that should take
9:41
care of
9:43
the yeah so that this should be good so
9:45
I told you this was going to be
#Mathematics
#Other