[music] How do I antidifferentiate trig functions?
Sine and cosine are easy if we just remember their derivatives.
The derivative of sine of x, with respect to x, is cosine x.
And the derivative of negative cosine is sine.
So what are their anti-derivatives? Knowing this, now I know some of the
anti-derivatives. If the derivative of sine is cosine, then
the anti-derivative of cosine is sine plus C.
And if the derivative of negative cosine is sine.
That means the anti-derivative of sine is negative cosine plus c.
Well what about tangent? What would I differentiate to get tangent
of x? Yeah, I'm asking what do I differentiate
to get tangent or what's the anti-derivative of tangent?
I'm claiming, that the antiderivative of tangent is log secant x, plus some
constant. How do I know this?
Well, to verify this is true, it's enough to do a differentiation problem.
I need to check that the derivative of log of secant of x Is equal to tan of x.
And how do I know this? Well I have to differentiate this
composition of functions. The composition of log and secant,.
Which I'll do in the chain rule, right. So what's the derivative of log?
It's one over. So it's 1 over the inside function times
the derivative of secant. What's the derivative of secant?
It's secant tangent. Alright, so this is 1 over secant times
secant x times tangent x. Now good news the secants cancel.
And what I'm left with is just tangent x. So I've found an antiderivative for
tangent. But it's totally opaque as to how somebody
would ever have gotten that as the anti-derivative of tangent.
And yet, once that candidate anti-derivative was revealed to us, we can
verify that it's truly an anti-derivative just by differentiating it and checking
that it really gives us tangent. Ok, let's try another one of these kinds
of problems. What's an anti-derivative of secant?
So I'm asking for a function, whose derivative is secant x.
Right? I'm looking for the antiderivative of
secant x. What is that?
Well it turns out that the antiderivative of secant x is this.
It's log, the absolute value of secant x plus tangent x.
Now again, it's totally mysterious, right, where this came from.
But we can verify that that's truly an antiderivative.
So to verify that the antiderivative of secant x is this, it's enough to
differentiate this and end up with secant x.
So let's try that, alright? I'll differentiate log of the absolute
value of secant x plus tangent x and what do I get?
Well, the derivative of log is 1 over the inside function, which is secant x plus
tangent x. Times the derivative of the inside
function. This is really the chain rule in action.
All right. Now, what's the derivative of this sum?
Well, that's the sum of derivatives, right?
So the derivative of secant is secant x tangent x.
And the derivative of, tangent. Is secant squared, and this is divided by
secant x plus tangent x. Now note I can factor out a secant x here,
and I've got secant x times the numerator is tangent x plus secant x.
The denominator's the same thing. I mean I wrote it in the opposite order.
But now this is great, right? This cancels and what I'm left with is
secant x. So yeah, the derivative of this is secant
x and consequently the antiderivative of secant x is this.
This is really a key in to anti multiplying, to factoring.
If you remember our experience, multiplying and anti-multiplying, or
factoring numbers, right. Factoring was hard.
I had to do a bunch of trial division to figure out how to factor a number.
But once I had that factorization, or purported factorization, in hand, then
checking it was easy. Multiplying is something that you can do.
It's the same story here with derivatives, right.
Anti differentiating is hard, I mean it seems hard to figure out what the anti
differentiating derivative is. But once you show me a guess, I can verify
that your guess is correct by doing the easy step of just differentiating and
seeing if you're right. I hope that these examples are giving you
some real respect for the true difficulties of anti-differentiation.
Differentiating is a mechanical process. You just have to apply the rules
carefully. But we've seen now that
anti-differentiation requires flashes of insight, just cooking up the correct
answer and then verifying that it's correct.
Well this seems, you know, impossible to continue, right?
How can we ever find anti-derivatives if we just have to keep hoping for insight to
appear? Well the good news is that most of the
rest of this course will be developing tools to make this kind of guessing more
systematic right? Instead of relying on just sheer insight
to solve these problems. We're going to be providing tools that
will aid your insight or aid your creativity, or provide a little bit of
structure so that you can exploit the patterns that we're going to be seeing,
and order the finite derivatives much more easily.
So I think it seems hard right now, but it's going to get a lot easier.