[Voiceover] Let's see if we can take the indefinite integral of cosine of X to the third power. I encourage you to pause the video and see if you can figure this out on your own. You have given it a go and you might have gotten stuck. Some of you all might have been able to figure it out, but some of you all might have gotten stuck. Find Used Opel Corsa Lite cars for sale with your local classified site Gumtree South Africa. Post FREE Ad Finish posting the ad you started! Opel corsa lite plus 1.4i 2006 model R47 000 Not NEGCar has low Mileage 124000.Car has no faults at all.Car is very neat and used daily from PMB to Camperdown.car is very lite on fuel 6L to 100KMCar. Opel corsa gsi. Free PDF Downloads for all Engine sizes and models for Opel Corsa. Toggle navigation. Opel Corsa Service and Repair Manuals. Every Manual available online - found by our community and shared for FREE. Workshop Manual. Opel Corsa is a mini car. 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Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p Formulas and Identities Tangent and Cotangent Identities sincos tancot cossin qq qq qq == Reciprocal Identities 11 cscsin sincsc 11 seccos cossec 11 cottan. How do I get Antares Autotune without an iLok? Find answers now! 1 Questions & Answers Place. We know that sin squared X plus cosine squared X is equal to one, or if we subtract sin squared from both sides, we know that cosine squared X is equal to one, write it this way, is equal to one minus sin squared X. What would happen if cosine to the third power, that's cosine squared times cosine. What happens if we were to take that cosine squared? Let me just rewrite it. This is the same thing as cosine of X times cosine squared of X, DX. What if we were to take this thing right over here, let me do that magenta color. What if we were to take this right over here and replace it with this. I now what you're thinking. 'Sal, what's that going to do for me? 'This feels like I'm making this integral 'even more convoluted.' What I would tell you, I would say, 'This might seem like it's getting more complicated, 'but as you explore and you play with it, 'you'll see that this actually makes 'the integral more solvable.' Let's try it out. If we do that, this is going to be equal to the indefinite integral, cosine of X times one minus sin squared X, DX. What is this going to be equal to? This is going to be equal to, let me do this in that green color. This is going to be equal to the indefinite integral of cosine X. I'm just going to distribute the cosine of X. Cosine of X minus, minus cosine of X, cosign of X sin squared of X, sin squared, sin squared X and then I can close the parentheses, DX. This, of course, is going to be equal to the integral of cosine of X, DX, and we know what that's going to be, minus the integral. I'll switch to one color now, of cosine of X, sin squared X, sin squared X, DX. Now, this is where it gets interesting. This part right over here is pretty straight forward. The anti-derivative of cosine of X is just sin of X. This right over here is going to be sin of X. I'll worry about the plus C at the end because both of these are going to have a plus C, so might as well just put one big plus c at the end. That's sin of X and then what do we have going on over here? Well, you might recognize, I have a function of sin of X. I'm taking sin of X and I'm squaring it and then I have sin of X's derivative right over here. This fits the, I have some derivative of a function and then I have another and then I have a, I guess you could say, a function of that function. That's a sin that maybe U substitution is in order, or we've seen the pattern, we've seen this show multiple times already, that you could just say, 'Okay, if I have a function of a function 'and I have that functions derivative, 'then essentially I can just take the anti-derivative 'with respect to this function.' This would be equal to, say, capital G is the anti-derivative of lower case G. Capital G of F of X plus C. Now, if what I said didn't make sense, then we could do U substitution and go through it a little bit more step by step. Let's just do that because we want things to make sense. That's the whole point of these videos. We could say U is equal to sin of X and then DU is going to be equal to cosine of X, DX. This part and that part is going to be DU and then this is going to be U squared. This is going to be minus. We have the integral of U squared, DU. What is this going to be? This is going to be, we're going to have negative U to the third power over three.
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