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let's see if we can evaluate the indefinite integral of tangent X DX and so 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 it's a reverse chain rule that's kind of you're seeing a function you think it's derivative and you can integrate with respect to that function all you see here or 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 its defined in terms of sine and cosine because we do at least have some tools and I at our disposal for dealing with sines and cosines or at least our brains at least my brain has an easier time processing them but we know tangent of X is the same thing as sine of X over cosine of X so let me rewrite it that way so this is equal to 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 you could even write it as sine of X times 1 over cosine of X so if you couldn't figure it out the first time I encourage you to pause the video again and once again think reverse chain reverse chain rule so what am I talking about when I keep saying reverse chain rule so let's let's just review that before I before I proceed with this example because if I were to just tell you well what's the indefinite integral of 1 over X DX well we know that that's going to be the natural log of the absolute value of X plus C now if I were to totally ask you what is the indefinite integral of f prime of x times 1 over f of X DX what is that going to be well here's where the reverse chain rule applies where I have 1 over f of X Y I would like if only I had its derivative being multiplied by this thing that I could just integrate with respect to f of X well I do I have its derivative right over here I felt applying times this thing so I could 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 that is on made it to whites you can't see everything but that's exactly what's going on right over here I have if I say that this thing right over here if cosine of X is f of X then sine of X is not quite the derivative it's the negative of the derivative so f prime of X would be negative sine of X so how do i how do I get that how do i engineer it well what if I just throw a negative there and a negative there so it's essentially multiplying by negative 1 twice which is still going to stay positive so now negative sine of X right over here I'm trying to squeeze it in between the integral sign 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 so this is f prime F prime of X so I can just apply the reverse chain rule this is going to be we deserve a little mini drumroll 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 and then of course we have our plus C and we can't forget 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 that we should know how to take the indefinite integral of the indefinite integral of tangent of X is and it's neat that these are connected in this way is the negative natural log of the absolute value of cosine of X plus C