Video Transcript
Calculate the definite integral
from 𝜋 over six to 𝜋 over three for the tangent of 𝑥 with respect to 𝑥.
To calculate this definite
integral, we’ll need to first remember what the integral of tangent 𝑥 d𝑥 would
be. The integral of the tangent 𝑥 d𝑥
equals the negative natural log of the absolute value of cosine of 𝑥. And that means we’ll need to
calculate the negative natural log of the absolute value of cosine of 𝑥 from 𝜋
over six to 𝜋 over three.
To do this, our first step is to
plug in 𝜋 over three. And we’ll have the negative natural
log of the cosine of 𝜋 over three. And we’ll subtract the negative
natural log of the absolute value of cosine of 𝜋 over six. We know that cosine of 𝜋 over
three equals one-half. So we’ll simplify our first term to
the negative natural log of one-half.
What about the cosine of 𝜋 over
six? That’s the square root of three
over two. So we then have the negative
natural log of the absolute value of the square root of three over two. This is important. Remember that we’re bringing down
the subtraction. And we recognize that this is as
subtracting a negative, which we’ll rewrite as addition. To simplify this a little further,
I noticed that we have a negative natural log of one-half. And we’re adding that to the
natural log of the square root of three over two.
We can change the order so that it
says the natural log of the absolute value of the square root of three over two
minus the natural log of one-half. We do this because we know the
general form — the natural log of 𝑎 minus the natural log of 𝑏 — equals the
natural log of 𝑎 over 𝑏. We can use this simplification to
get the natural log of the square root of three over two over one-half. This is saying the square root of
three over two divided by one-half, which is the same thing as the square root of
three over two times two over one. And that simplifies to the square
root of three over one or simply the square root of three. We’ve simplified this down to be
the natural log of the absolute value of the square root of three.
This is one way to write this
definite integral. However, there might be times when
we would be interested in getting rid of the square root. To get rid of the square root,
we’ll need to consider another way to write the square root of three we know that
the square root of three equals three to the one-half power. We can write this natural log as
the natural log of three to the one-half power. This is important because we know
that the natural log of 𝑎 to the 𝑏 power equals 𝑏 times the natural log of
𝑎.
We can take the exponent, the
one-half, and bring it out of the natural log and multiply that by the natural log
of three. Another form of the natural log of
the square root of three is to say the natural log of three over two.
