In this video, we're gonna go over
the derivatives and and the integrals
of the six basic trig functions.
So let's take a look at them.
We'll start with derivatives,
'cause derivatives are easier.
And of course the derivative of
sin(x) is cos(x),
and the derivative of cos(x) is -sin(x).
And those you should have memorized
down in your bones.
And then the other four trig functions,
if you want the derivatives of them.
Well they're just ratios
of sines and cosines,
and so you get them by the quotient rule.
And the derivative of tan(x)
winds up being sec^2(x),
the derivative of sec(x) winds up
being sec(x)tan(x),
and those come up a lot.
We will find these formulas to
be very, very useful.
Now, the derivative of all the
co-functions
just looks like the derivative of the
original functions, with a sign shift.
If the derivative of sin(x) is cos(x),
the derivative of cos(x) is...
co-cos(x), that's sin(x),
but with a minus sign.
Derivative of cot(x) looks just like
the derivative of tan(x),
except instead of sec^2(x),
you get csc^2(x),
and you get a minus sign.
The derivative of csc(x) looks just
like the derivative of sec(x),
except instead of getting sec(x),
you get csc(x),
instead of tan(x), you get cot(x),
and you get a minus sign.
So that's it for derivatives.
So now let's start figuring out integrals.
The first two integrals are really easy.
You know the integral
of sin(x) is -cos(x),
because the derivative
of -cos(x) is sin(x).
And the integral of cos(x) is sin(x).
So those are really easy,
we'll just write them down.
Integral of sin(x)dx is -cos(x) + C,
integral of cos(x)dx is +sin(x) + C.
The first interesting case is tangent.
So the way we do tangent is when
in doubt, write everything
in terms of sines and cosines.
And now you notice, hey, the numerator is
minus the derivative of the denominator.
So this wants to be done
with a u-substitution,
where u = cos(x),
and then du is -sin(x)dx.
So this becomes the integral of -du/u,
which is -ln|u| + C.
So that's -ln|cos(x)| + C.
Now you'll sometimes see this
written as ln|sec(x)| + C.
The reason is the secant is
the reciprocal of the cosine.
So ln|sec(x)| is -ln|cos(x)|.
So you can write it this way,
you can write it that way,
it's the same function.
Next, we've got cotangent.
Well this is the integral of
cos(x)/sin(x),
and now we do a u-substitution
where u = sin(x).
So du = cos(x)dx,
and this just gives us integral of du/u.
And that is ln|u| + C,
so you can write that as
ln|sin(x)| + C,
or you could write it as
-ln|csc(x)| + C.
All of those are equivalent.
Now come the hard cases.
The hard cases are secant
and cosecant.
So secant involves a really sneaky trick,
the kind of trick that you wouldn't
think of in a million years,
but fortunately somebody really clever
about 350 years ago thought of the trick,
and we're gonna piggy back
off of what he did.
See what we're gonna do is we're
gonna multiply sec(x) by 1.
By 1 in a fancy way.
We're gonna multiply by
(sec(x) + tan(x))/(sec(x) + tan(x)).
I think we can all agree that that's one.
And that becomes the
integral of sec^2(x) + sec(x)tan(x)
divided by sec(x) + tan(x).
And now remember our table.
The derivative of sec(x) is sec(x)tan(x).
The derivative of tan(x) is sec^2(x).
So the derivative of the
denominator is the numerator.
The derivative of the sec(x)
is sec(x)tan(x),
the derivative of tan(x) is sec^2(x).
So this is the integral of du/u,
where u is sec(x) + tan(x).
And that just gives us
ln|sec(x) + tan(x)| + C.
Now for the cosecant,
it's the same process.
We write this as the integral of
csc(x) times csc(x) + cot(x)
divided by csc(x) + cot(x),
and that winds up being
-ln|csc(x) + cot(x)|.
Plus a constant of course.
'Cause the derivative of csc(x)
is -csc(x)cot(x),
the derivative of cot(x) is -csc^2(x).
Okay, so let's summarize.
We've got that the derivative of
sin(x) is cos(x),
and the integral of sin(x)
is -cos(x) + C.
The derivative of cos(x) is -sin(x),
the integral of cos(x) is sin(x) + C.
Watch out for these minuses.
You'll find amazingly many mistakes
where people say the integral
of sin(x) is cos(x) instead of -cos(x).
Or the derivative of cos(x)
is sin(x) instead of -sin(x).
Careful about the pluses and minuses.
The derivative of tan(x) is sec^2(x),
the integral of tan(x) is -ln|cos(x)| + C,
the derivative of cot(x) is -csc^2(x),
its integral is ln|sin(x)| + C.
Derivative of sec(x) is sec(x)tan(x),
integral of sec(x) is
ln|sec(x) + tan(x)| + C,
derivative of csc(x) is -csc(x)cot(x),
and the integral is
-ln|csc(x) + cot(x).
For all of these formulas,
for the co-functions,
if you know what it is
for sin(x), tan(x), and sec(x),
you automatically can figure out
what it is for cos(x), cot(x), and csc(x).
It's the same formulas, only with
the co-functions, and with a minus sign.
It works for derivatives,
it works for integrals as well.
