Video Transcript
The integral between two and ∞ of
𝑒 to the negative five 𝑝 with respect to 𝑝 is convergent. What does it converge to?
Because this integral has an
infinite limit, we call this an improper integral. We have a general result that tells
us that the integral between 𝑎 and ∞ of 𝑓 of 𝑥 with respect to 𝑥 is the limit as
𝑡 approaches ∞ of the integral between 𝑎 and 𝑡 of 𝑓 of 𝑥 with respect to
𝑥. And we know that this integral is
convergent, which means the integral must approach a specific value. And that’s what we’re going to
find.
What this result tells us is that
we can replace the infinite limit in our integral with a variable 𝑡 and then take
the limit as 𝑡 approaches ∞ of our integral. So we’re going to find the limit as
𝑡 approaches ∞ of the integral between two and 𝑡 of 𝑒 to the power of negative
five 𝑝 with respect to 𝑝. To do this, we recall the general
rule that the integral of 𝑒 to the 𝑘𝑥 power with respect to 𝑥 is equal to one
over 𝑘 multiplied by 𝑒 to the 𝑘𝑥 power plus a constant of integration 𝑐. So the integral of 𝑒 to the power
of negative five 𝑝 with respect to 𝑝 is one over negative five 𝑒 to the power of
negative five 𝑝.
Note that because we have limits of
integration for this question, we don’t need to include a constant of
integration. Let’s write this as a single
faction, negative 𝑒 to the power of negative five 𝑝 over five. And now, let’s apply these
limits. And when we do this, we have to
remember that our integral is negative. So when we apply the limits, we’re
going to be subtracting a negative. So we can replace this with an
add. And we can replace negative five
multiplied by two with negative 10. But we still need to apply this
limit.
If we rewrite the term negative 𝑒
to the power of negative five 𝑡 over five, using the fact that 𝑒 to the negative
five 𝑡 is the same as one over 𝑒 to the five 𝑡, and then we recognize that we can
simplify this to negative one over five 𝑒 to the five 𝑡. And we know the fact that the limit
as 𝑡 approaches ∞ of 𝑒 to the 𝑡 power is ∞. So the denominator approaches
∞. So the term approaches zero. So in fact, when we apply the limit
to this bracket, what we have left is 𝑒 to the negative 10 power over five, which
shows that this limit approaches a specific value just as we expected.
