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Integrals involving Exponential Functions Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Integrals involving Exponential Functions Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. How do you integrate exponential functions? |  |
| 2. Can you provide an example of integrating an exponential function? | |
Ans. Sure! Let's integrate the function ∫2e^x dx. Using the formula, we have:
∫2e^x dx = 2∫e^x dx = 2(e^x) + C
So, the integral of 2e^x is 2e^x + C.
| 3. Is there a specific method to integrate exponential functions with a constant as the base? | |
Ans. Yes, if you have an exponential function with a constant base, such as ∫a^x dx, you can use a logarithmic transformation. The integral formula is:
∫a^x dx = (a^x / ln(a)) + C
where ln(a) represents the natural logarithm of the base a.
| 4. Can you explain how to integrate a function that is the product of an exponential function and another function? | |
Ans. Yes, to integrate a function of the form ∫f(x)e^x dx, you can use integration by parts. The formula is:
∫u dv = uv - ∫v du
Let u be the function f(x) and dv be e^x dx. Differentiate u to find du and integrate dv to find v. Then, apply the integration by parts formula to evaluate the integral.
| 5. Are there any special techniques for integrating more complex exponential functions? | |
Ans. Yes, some techniques for integrating more complex exponential functions include substitution and partial fraction decomposition. These methods can be helpful when the integrand involves a combination of exponential functions, trigonometric functions, or other algebraic terms. It's important to analyze the specific form of the integral and choose the most appropriate technique based on its structure.
Text Transcript from Video
hi guys we're going to be covering an
integration of exponential functions in
this session let's get started now if
you want to integrate e of f of X DX
then when you integrate it you would get
f of f of X divided up divided by f dash
of X plus C now you kind of will start
seeing similarities between this and
differentiation and differentiation you
would actually differentiate the power
and multiply in the front for
integration you differentiate the power
and you divide it so that's something
that you just need to remember now let's
have a look at a couple of examples
let's see if we can get this idea
working is three examples guys we've got
integral of e to the power of 4x
integral of e to the power of 7 X minus
2 and finally 2 divided by e to the
power of 3x ok starting on the first one
we write the function as it is so it's e
to the power of 4x and then we divide it
by the integrated integral not integral
sorry we divide it by the
differentiation of the power so see how
we've got 4x here when you differentiate
4x you get 4 so therefore you divide it
by 4 if you want to integrate it of
course whenever you integrate anything
you got to have the plus C which is the
constant the arbitrary constant which we
don't know which was there or not so
moving on to the next one
once again we write the function as it
as it is e to the power of 7 X minus 2
and that's divided by the
differentiation of the power so if we
have a look
we've got 7x minus 2 if we differentiate
7x minus 2 we would get 7 therefore u
divided by 7 and once again don't forget
the plus C okay next example now in this
case before we start it'll be good to
put the e to the power of 3x in the
numerator so you would get 2 e negative
3x DX now once we start integrating it
we write down the
function as it is so we're going to have
to e negative 3x and that's divided by
the differentiation of the power the
power in this case is negative 3x if we
differentiate that function we will get
negative 3 and of course don't forget
the plus C now guys in integration
particularly in our external you can
actually leave the answer at this point
but I still encourage people to simplify
it further because it just strengthens
your algebra skills so if I want to go
further from here I would have it as
negative to a negative 3x divided by 3
plus C and of course the last step would
be to bring e to the power of negative 3
X back to the denominator as a positive
power so I would get negative 2/3 e
positive 3 X plus C ok now in the next
slide I'll be looking at a slightly
different type of integration but let's
have a look ok so in this example I want
to be able to integrate e to the power
of 3x plus 1 divided by e to the power
of X now the first step that I would do
in this case is separate them as two
fractions so I would have e to the power
of 3x divided by e to the power of X
plus 1 divided by e to the power of X DX
now I can start canceling things out a
bit and making things a little bit more
easier for me
so here I would get e to the power of 2x
as well 3x take away take away X 1 X
would be 2x and on this side because
it's e to the power of X is in the
denominator I could bring it to the
numerator and make it e to the power
negative X DX
now I can actually start integrating it
so when I integrate e to the power of 2
X I would write the function as it is
and I would divide by the
differentiation of the power so here
the powers well the function is to X if
I differentiate to X I get two plus I
write e to the power of negative x as it
is and I differentiate the power or the
function on top of e so this is in the
purple here if I differentiate negative
x I get negative one and of course
because I am integrating I shouldn't
forget the C okay at this point again
you can leave it here you'd still be
correct but I'm going to continue
simplifying further on now the first one
there's nothing really I can do with the
first fraction that'll just be e to the
2x divided by two
but the next one will become minus
because of the negative one minus e to
the power of negative x plus C and one
more line guys I want to bring eetu the
power of negative x to the denominator
which makes it 1 divided by e to the
power of X plus C alright that's it for
integrating exponential functions thanks
for watching
integration of exponential functions in
this session let's get started now if
you want to integrate e of f of X DX
then when you integrate it you would get
f of f of X divided up divided by f dash
of X plus C now you kind of will start
seeing similarities between this and
differentiation and differentiation you
would actually differentiate the power
and multiply in the front for
integration you differentiate the power
and you divide it so that's something
that you just need to remember now let's
have a look at a couple of examples
let's see if we can get this idea
working is three examples guys we've got
integral of e to the power of 4x
integral of e to the power of 7 X minus
2 and finally 2 divided by e to the
power of 3x ok starting on the first one
we write the function as it is so it's e
to the power of 4x and then we divide it
by the integrated integral not integral
sorry we divide it by the
differentiation of the power so see how
we've got 4x here when you differentiate
4x you get 4 so therefore you divide it
by 4 if you want to integrate it of
course whenever you integrate anything
you got to have the plus C which is the
constant the arbitrary constant which we
don't know which was there or not so
moving on to the next one
once again we write the function as it
as it is e to the power of 7 X minus 2
and that's divided by the
differentiation of the power so if we
have a look
we've got 7x minus 2 if we differentiate
7x minus 2 we would get 7 therefore u
divided by 7 and once again don't forget
the plus C okay next example now in this
case before we start it'll be good to
put the e to the power of 3x in the
numerator so you would get 2 e negative
3x DX now once we start integrating it
we write down the
function as it is so we're going to have
to e negative 3x and that's divided by
the differentiation of the power the
power in this case is negative 3x if we
differentiate that function we will get
negative 3 and of course don't forget
the plus C now guys in integration
particularly in our external you can
actually leave the answer at this point
but I still encourage people to simplify
it further because it just strengthens
your algebra skills so if I want to go
further from here I would have it as
negative to a negative 3x divided by 3
plus C and of course the last step would
be to bring e to the power of negative 3
X back to the denominator as a positive
power so I would get negative 2/3 e
positive 3 X plus C ok now in the next
slide I'll be looking at a slightly
different type of integration but let's
have a look ok so in this example I want
to be able to integrate e to the power
of 3x plus 1 divided by e to the power
of X now the first step that I would do
in this case is separate them as two
fractions so I would have e to the power
of 3x divided by e to the power of X
plus 1 divided by e to the power of X DX
now I can start canceling things out a
bit and making things a little bit more
easier for me
so here I would get e to the power of 2x
as well 3x take away take away X 1 X
would be 2x and on this side because
it's e to the power of X is in the
denominator I could bring it to the
numerator and make it e to the power
negative X DX
now I can actually start integrating it
so when I integrate e to the power of 2
X I would write the function as it is
and I would divide by the
differentiation of the power so here
the powers well the function is to X if
I differentiate to X I get two plus I
write e to the power of negative x as it
is and I differentiate the power or the
function on top of e so this is in the
purple here if I differentiate negative
x I get negative one and of course
because I am integrating I shouldn't
forget the C okay at this point again
you can leave it here you'd still be
correct but I'm going to continue
simplifying further on now the first one
there's nothing really I can do with the
first fraction that'll just be e to the
2x divided by two
but the next one will become minus
because of the negative one minus e to
the power of negative x plus C and one
more line guys I want to bring eetu the
power of negative x to the denominator
which makes it 1 divided by e to the
power of X plus C alright that's it for
integrating exponential functions thanks
for watching
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