We're asked to consider the function f[x],
and that big F[x] be the antiderivative of f[x],
with big F[1] equal to zero.
Determine big F[x].
So, we'll first determine the indefinite integral of f[x]
which will give us the antiderivative
in the form of big F[x] plus C.
Then we'll determine the specific antiderivative
by determining the value of C,
since we know a big F[1] is equal to zero.
So, we begin by determining the indefinite integral
of five divided by x cubed minus two
divided by x to the sixth, where the variable of integration
is x indicated by differential x.
However, before we can integrate,
we to apply the negative exponent properties shown above
in order to apply the power rule of integration
shown here below.
Since one divided by x to the power of n
is equal to x to the power of negative n,
we can rewrite five divided by x to the third
as five x to the power of negative three.
And then we have minus two divided by x to the sixth
is equivalent to two x to the power of negative six.
And now, we can apply the power rule of integration
to find the general antiderivative.
The integral of five x to the power of negative three
is equal to five times the integral of x
to the power of negative three,
which is x to the power of negative three plus one
divided by the quantity negative three plus one,
and then minus the integral of two x
to the power of negative six,
which is equal to two times the integral of x
to the power of negative six,
which is x the power of negative six plus one
divided by the quantity negative six plus one.
And then we have plus C, any constant.
Let's go ahead and simplify.
We have five times x to the power of negative two
divided by negative two minus two times x
to the power of negative five,
divided by negative five plus C.
Let's continue simplifying.
The first term is negative 5/2 x
to the power of negative two.
Next we have minus two, and then we have a negative five
in the denominator, which simplifies
the plus 2/5 x to the power of negative five, plus C.
And then finally, let's rewrite this
using positive exponents.
So, we have the general antiderivative big F[x]
is equal to negative five divided by the product
of two index squared, plus two divided by the product
of five and x to the fifth, plus C.
So, this is the first step in the process.
We now have the general antiderivative,
but now, since we know big F[1] is equal to zero,
we can determine the value of C
to determine the specific antiderivative,
and let's do this on the next slide.
Since we know big F[1] is equal to zero,
when we substitute one for x,
the function value must equal zero,
which gives us the equation zero equals negative five
divided by the product of two and one squared plus two
divided by the product of five and one to the fifth,
and then we have plus C.
And now we solve the equation for C.
We have zero equals negative 5/2,
plus 2/5, plus C.
Let's add these two fractions
by obtaining a common denominator, which would be 10.
We have zero equals negative 25/10 plus 4/10,
which is negative 21/10, plus C.
And now, to solve for C we added 21/10 to both sides,
which gives us 21/10 equals C.
And therefore the specific antiderivative we are looking for
is big F[x] is equal to negative five
divided by two x squared, plus two
divided by five x to the fifth.
And then since C is 21/10, we have plus 21/10.
I hope you found this helpful.