We want to evaluate the given definite integral
using integration by parts, which is this formula here,
which states the integral of u dv
is equal to u v minus the integral of v du.
The idea here is that this integral here,
the integral of u dv is difficult to integrate.
But if we apply this formula,
the integral of v du will be easier to evaluate.
Now looking at this integral here,
just because u is first does not mean
u will be equal to the first part of the integrand,
and the second part will be equal to dv.
We want to select the u so that du is simpler.
We want to select dv so we can easily integrate to find v.
So in this example, let's let u
be equal to natural log three t,
and therefore dv is equal to t to the fourth dt.
Now for some reason, if we apply integration by parts
and this integral here is not easier to integrate,
we could always come back and try something different.
So again, first u is equal natural log three t.
And therefore differential u is equal to u prime times dt.
So differential u is equal to,
the derivative of natural log three t
would be equal to one over three t
times the derivative of three t,
which would be three, and then times dt.
And notice how this simplifies to just one over t dt.
So differential u is one divided by t dt.
And dv is equal to t to the fourth dt.
So now we integrate to find v,
which would give us t to the fifth divided by five,
or 1/5 t to the fifth.
And now we can apply the integration by parts formula.
So the given integral is equal to u times v.
So we'd have natural log three t times 1/5 t to the fifth,
or 1/5 t to the fifth natural log three t.
And notice how we do have an definite integral here.
So we need to evaluate this at the upper
and lower limits of integration
and then find the difference.
And we still have minus the integral of v du.
Well here's v and here's du.
So we'd have 1/5 t to the fifth times
one divided by t dt.
The limits of integration are still from four to seven.
Let's go ahead and simplify this.
We can simplify one factor of t.
Let's also factor out the 1/5.
We'll evaluate this a little bit later.
So we have minus 1/5 times the integral of,
this would be again t to the fourth dt.
Now we can go ahead and integrate.
We'd have minus 1/5 times t to the fifth divided by five.
So we'd have 1/5 times 1/5 t to the fifth,
or minus 1/25 t to the fifth.
And now we'll evaluate this at seven,
then four, then find the difference.
Let's go ahead and do this on the next slide.
So when t is seven, we'd have 1/5 times seven to the fifth
times natural log three times seven, that's 21,
minus 1/25 times seven to the fifth.
And then when t is four, we'd have 1/5 four to the fifth,
natural log three times four, that'd be 12,
minus 1/25 four to the fifth.
Now we'll go to the calculator.
I do have the newest version of the TI-Smart View
so when I enter these expressions,
it will look slightly different.
Let's evaluate this first when t is seven.
So we'd have 1/5
times seven raised to the fifth.
And to get out of the exponent,
I'm going to press the right arrow.
And then we have natural log 21
minus 1/25
times seven to the fifth.
Enter.
So here we have approximately 9561.5777.
And now when t is four, I'm going to press 2nd + Enter,
which allows me to edit the previous entry.
So I'll scroll back to the very beginning
and I'll change the seven to base four,
and I'll change natural log 21 to natural log 12,
and change
this base of seven to base four.
Enter.
And so we have approximately 467.9489.
And now we'll find the difference.
So we have 9561.5777
and then minus 467.9489.
Enter.
So we have approximately 9093.6288.
And now of course because we do have a definite integral,
we can check the entire integral
using the graphing calculator.
So let's go ahead and do that.
Just keep in mind this is a rounded value,
so the exact value might be slightly different.
If we press Math and then
go down to function integration
which I believe is option nine, which it is, press enter.
And again this looks different than previous videos
because this is the newest operating system.
So going back to the original problem,
the lower limit of integration is four, right arrow.
Upper limit of integration is seven, right arrow.
The integrand is t to the fourth natural log three t,
but we'll use x instead of t.
So we have x raised to the power of four, right arrow,
natural log three x,
close parenthesis, right arrow,
the variable of integration, which is x.
And enter.
So going back to our answer,
notice how this verifies our answer is correct.
I hope you found this helpful.