Integral of tan x (video) | Khan Academy

Video transcript

– [ Voiceover ] Let ‘s see if we can evaluate the indefinite built-in of tangent x dx. Like constantly, pause the video and see if you can figure it out on your own and I will give you a touch, 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 derived function and you can integrate with deference to that function. All you see here is a tangent of ten, 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 administration 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 ten is the like thing as sine of ten over cosine of x so let me rewrite it that way. This is equal to the indefinite integral of sine of ten over cosine of x dx and you could even write it this way and this is a little morsel of a tip. You could even write it as sine of x times one over cosine of x. If you could n’t figure it out the first gear time, I encourage you to pause the video again and, once again, think reverse chain rule. then, what am I talking about when I keep saying reverse chain rule ? Let ‘s barely review that before I proceed with this exercise. If I were to just tell you, well what ‘s the indefinite integral of one over ten dx ? We know that, that ‘s going to be the natural log of the absolute measure of x plus c. now, if I were to ask you what is the indefinite built-in of fluorine prime of x times one over f of x dx. What is that going to be ? hera ‘s where the reverse chain predominate applies. Where I have one over fluorine of x, if only I had it ‘s derivative being multiplied by this thing then I could precisely 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 invert 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 fluorine of x plus degree centigrade and that is precisely … I ‘ve made it besides wide-eyed so you ca n’t see everything … but that ‘s precisely what’s going on correct over here. I have, if I say this thing right over here, cosine of x is degree fahrenheit of adam, then sine of x is not quite the derived function, it ‘s the damaging of the derivative instrument. 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 veto there and a veto there so it is basically multiply 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 built-in sine and the sine of ten, this right over here, immediately that I put a damaging sine of ten, that is the derivative of cosine of x. This is f flower of x so I can barely apply the overrule chain principle. This is going to be, we deserve a little mini drum peal here, this is going to be equal to the natural log of the absolute value of our degree fahrenheit of x, which is going to be cosine of x. then, of run, we have our plus coulomb and we ca n’t forget we had this little negative sitting out here so we ‘re going to have to put the minus correct over there. And we are done, we equitable figured out that ‘s kind of a neat consequence because it feels like that ‘s something we should know how to take the indefinite built-in of. The indefinite integral of tangent of ten is, and it ‘s neat they ‘re connected in this way, is the minus natural log of the absolute rate of cosine of ten plus c.

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