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## Reverse chain rule

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# Integral of tan x

## Video transcript

- [Voiceover] Let's see if we can evaluate the indefinite integral of tangent x dx. Like always, pause the video and see if you can figure it out on your own and I will give you a hint,
think reverse chain rule. Alright, so you have attempted it and you would say well reverse chain rule, that's kind of your seeing a function and your seeing it's derivative and you can integrate with
respect to that function. All you see here is a tangent
of x, what am I talking about? Well, whenever you see a tangent of x or a cosecant or a secant,
at least in my brain, I always like to break it
down into how it's defined in terms of sine and cosine because we do at least have some tools at our disposal for dealing with sines and cosines. Or at least our brains, at least my brain, has an easier time processing them. We know tangent of x is
the same thing as sine of x over cosine of x so let
me rewrite it that way. This is equal to the indefinite
integral of sine of x over cosine of x dx and you
could even write it this way and this is a little bit of a hint. You could even write it as sine of x times one over cosine of x. If you couldn't figure
it out the first time, I encourage you to pause the video again and, once again, think reverse chain rule. So, what am I talking about when I keep saying reverse chain rule? Let's just review that before
I proceed with this example. If I were to just tell you, well what's the indefinite integral of one over x dx? We know that, that's going to be the natural log of the
absolute value of x plus c. Now, if I were to ask you what is the indefinite integral of f prime of x times one over f of x dx. What is that going to be? Here's where the reverse
chain rule applies. Where I have one over f of x,
if only I had it's derivative being multiplied by this
thing then I could just integrate with respect to f of x. Well, I do, I have it's
derivative right over here. It's multiplying times this thing. I can use the reverse chain rule to say that this is going to be equal to the natural log of the absolute value of the thing that I
have in the denominator, which is f of x plus c
and that is exactly... I've made it too wide so
you can't see everything... but that's exactly what's
going on right over here. I have, if I say this
thing right over here, cosine of x is f of x, then sine of x is not quite the derivative, it's the
negative of the derivative. F prime of x would be negative sine of x. How do I get that, how do I engineer it? What if I just threw a negative there and a negative there so it is essentially multiplying by negative one twice which is still going to stay positive. Negative sine of x, right over here, I'm trying to squeeze it in
between the integral sine and the sine of x, this right over here, now that I put a negative sine of x, that is the derivative of cosine of x. This is f prime of x so I can just apply the reverse chain rule. This is going to be, we deserve a little mini drum roll
here, this is going to be equal to the natural
log of the absolute value of our f of x, which is
going to be cosine of x. Then, of course, we have our plus c and we can't forget we
had this little negative sitting out here so we're going to have to put the
negative right over there. And we are done, we just figured out that's kind of a neat result because it feels like that's something we should know how to take
the indefinite integral of. The indefinite integral
of tangent of x is, and it's neat they're connected
in this way, is the negative natural log of the absolute
value of cosine of x plus c.