### Video Transcript

Calculate the integral from
negative one to 𝑒 minus two of five over 𝑥 plus two with respect to 𝑥.

In this question, we’re looking to
evaluate an integral between the limits of negative one and 𝑒 minus two. We’ll begin then by integrating
five over 𝑥 plus two with respect to 𝑥. Now whilst it might look like we
need to perform a substitution to be able to complete this integral, there is in
fact a standard result that we can quote. We can integrate one over 𝑎𝑥 plus
𝑏 with respect to 𝑥 for constants 𝑎 and 𝑏. And we get one over 𝑎 times ln of
𝑎𝑥 plus 𝑏 plus the constant of integration 𝑐. Remember we can also take a
constant multiple outside of the integral and that allows us to focus on integrating
the function in 𝑥.

So we rewrite our integral as five
times the integral of one over 𝑥 plus two with respect to 𝑥. And now we can see that the
integral of one over 𝑥 plus two with respect to 𝑥 is ln of 𝑥 plus two. And since we’re evaluating this
between the limits of negative one and 𝑒 minus two, we don’t need to worry about
that constant of integration.

We now substitute 𝑒 minus two and
negative one into our integral. And it becomes five times ln of 𝑒
minus two plus two minus ln of negative one plus two. This simplifies to five times ln of
𝑒 minus ln of one. But remember ln of 𝑒 is simply one
and ln of one is zero. So we get five times one which is
equal to five.

And so, we see that the integral
between the limits of negative one and 𝑒 minus two of five over 𝑥 plus two with
respect to 𝑥 is five.