Welcome to a lesson on the
second fundamental theorem of calculus.
I've also seen this called part one
of the fundamental theorem of calculus in some text books.
The second fundamental theorem of calculus
states that if f is continuous on
an open interval I containing the constant a,
then for every x in the interval
the derivative with respect to x
of the integral of f of t from a to x
equals f of x.
So notice how to find the derivative of this integral
with a lower limit of integration is a constant a
and the upper limit of integration is the variable x
this is equal to the integrated function
where we substitute x for t,
giving us f of x.
We can think of this integral here
as the accumulation function a of x
where a of x would give us the area
of this shaded region where a is a constant
and x is any value to the right of a.
So the derivative with respect to x
of this accumulation function is just f of x.
The integrated function evaluated at x.
Now I do want to mention that if this
upper limit of integration is not x,
lets say it was two x or x squared,
it wouldn't be quite this straight forward.
We would have to apply the chain rule in
order to find the derivative of this integral.
I do have several examples of this in other videos.
But for our next step let's try to justify
this outcome before we take the shortcut.
One way to justify this would be to first find
this diff integral and then find
the derivative with respect to x.
So if we wanted to evaluate this diff integral
we would first find the anti-derivative
function big f of t.
So we'd have the derivative with respect to x,
of big f of t which we then evaluated x,
then a, then find the difference.
So we'd have the derivative with respect to x,
of big f of x, minus big f of a.
Remember a is a constant.
So now when we find the derivative with respect to x
this would give us big f prime of x
then the derivative of big f of a would be zero
since big f of a would be a constant.
And since big f is the anti-derivative of little f,
big f prime of x does equal f of x which is
given by the second fundamental theorem of calculus.
Lets go ahead and try this with the simple
integral as we see here.
Again before we take the shortcut given by
the theorem let's look at this the long way
and first evaluate this diff integral.
So we have the derivative with respect to x,
well the anti-derivative two t would be
two times t to the second divided by two.
Of course this simplified to just t squared.
So we'd have the derivative with respect to x
and then we have our anti-derivative t squared
which we need to evaluate x then a
and then find the difference.
So this would give us the derivative with respect to x of,
we substitute x for t we'd have x squared,
we substitute a for t we'd have minus a squared.
Now to find the derivative with respect to x,
the derivative of x squared would be two x,
and the derivative of a squared would be zero
since a squared is a constant.
So now looking back at this integral here,
notice that f of t in the integrated function
is two t and if we evaluate this integrated function
at x we'd have f of x equals two x
which is what we found the long way.
Now let's look at two examples and take the short-cut.
We have the derivative with respect to x of the integral
integrated from two to x,
so because the lower limit of integration
is the constant two,
and the upper limit of integration is x
to find the derivative of the integral
we just need to substitute x for t
into the integrated function.
So this is going to be equal to sin cubed of x squared
divided by the cube root
of x to the fifth plus one.
And for our last example,
again we have the derivative with
respect to x of this integral integrated from five to x
so applying the second fundamental theorem of calculus,
we can simply substitute x for t into the
integrated function to find
the derivative of this integral.
So we'd have 15 x raised to the power of 5/3
divided by natural log of x to the third
plus x to the second.
So again, the second fundamental theorem of calculus
shows us a relationship between
differentiation and integration and also
allows us to find the derivative of integrals
without finding the anti-derivatives.
And again, I do have several other videos
on this topic where we have to apply the chain rule
when applying the second fundamental theorem of calculus.
I hope you found this helpful.