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Current time:0:00Total duration:8:57

Video transcript

let's see if we can take the integral of cosine of 5x over e to the sine of 5x DX and there's a crow squawking outside of my window so I'll try to stay focused so let's think about whether u substitution might be appropriate your first temptation might decide hey maybe we let u equal sine of 5x and if u is equal to sine of 5x we have something that's pretty close to D u up here let's verify that so D you could be equal to so D u DX derivative of U with respect to X well we just use the chain rule derivative of 5x is 5 times the derivative of sine of five x with respect to 5x that's just going to be cosine of 5x if we want to write this in differential form which is useful when we do our u substitution we could say the D U is equal to 5 cosine 5 X now when we look over here we don't have quite d u there we have just cosine of 5x DX oh sorry I need cosine of 5x DX just like that so when you look over here you have a cosine of 5x DX but we don't have a 5 cosine of 5x DX but we know how to solve that we can multiply by we can multiply by 5 and divide by 5 1/5 times 5 is just going to be 1 so we haven't changed the value of the expression but when we do it this way we see pretty clearly we have our u we have our U and we have our D u our D U is 5 let me circle that let me do that in that blue color is 5 cosine of 5x DX so we can rewrite this entire expression as do that 1/5 in purple this is going to be equal to 1/5 I hope you don't hear that crow outside he's getting quite obnoxious 1/5 times the integral of all the stuff in blue is my deu all this stuff in blue is my D U and then that is over e to the u e to the U so how do we take the antiderivative of this well you might be tempted to well what would you do here well we're still not quite ready to simply take the antiderivative here if I were to rewrite this I could rewrite this as this is equal to 1/5 times the integral of e to the negative u e to the negative u du u e to the negative UD u and so what might jump out at you is maybe we do another substitution we've already used the letter U so maybe we'll use W we'll do some W substitution and you might be able to do this in your head but we'll do W substitution just to make a little bit clearer so let's this would been really useful if this was just either you because we know the antiderivative of e to the U is just e to use so let's try to get it in terms of the form of e to the something and not e to the negative something so let's set let's end I'm running out of colors here let's set W let's set W is equal to negative u in that case then D W derivative of W with respect to U is negative 1 or if we were to write that statement in differential form DW is equal to D u times negative 1 is negative negative D U so this right over here would be our W and do we have a DW here well we have just a D U we don't have a negative D U there but we can create a negative D U by multiplying this inside by a negative 1 but then also multiplying the outside by a negative 1 negative 1 times negative 1 is positive 1 we haven't changed the value we have to do both of these in order for it to make sense or I could do it like this so negative 1 over here and a negative 1 right over there and if we do it in that form then this negative 1 times D U that's the same thing as negative D U this this is this right over here and so we can rewrite our integral it's going to be equal to now it's going to be negative 1/5 negative 1/5 trying to use the colors as best as I can negative 1/5 times the indefinite integral of e to the well instead of negative view we can write w e to the W e to the W instead of D u times negative 1 or negative d u we can write DW d DW now this simplifies things a good bit we know what the antiderivative of this is in terms of W this is going to be equal to this is going to be equal to negative one-fifth negative one-fifth e to the W e to the W and then we might have some constant there so I'll just do a plus C and now we just have to do all of our unsubstituted so we know that W is equal to negative u so we could write that so this is equal to negative one-fifth I want to stay true to my colors negative one-fifth e to the negative u e to the negative u that's what W is equal to plus C we're still not done unhhhh substituting we know that U is equal to sine of 5x U is equal to sine of 5x so we can write this as being equal to negative one-fifth negative one-fifth times e times e to the negative U which is negative U is sine of 5x and then finally we have our we have our plus C now there was a simpler way that we could have done this by just doing one substitution but then you would have kind of had to look ahead a little bit and realize that it would have been not trivial to take you're not too bad to take your antiderivative of e to the negative u the insight that you might have had although you shouldn't really hold yourself if you feel too bad if you didn't see that insight is we could have rewritten that original integral we could have written the original integral let me rewrite it's cosine of 5x over e of e to the sine of 5x DX we could have written this entire original integral as being equal to cosine of 5x times e to the negative sine of 5x DX and in this situation we could have set u to be equal to negative 5x and say well if u is equal to or negative sine of 5x if u is equal to negative sine of 5x then D U is going to be equal to negative 5 cosine of 5x and we don't have a negative 500 DX we don't have a negative 5 here but we could construct one by putting a negative 5 there and then multiplying by negative 1/5 and then that would have immediately simplified this integral right over here to be equal to negative 1/5 x times the integral of well we have our D u Rd you let me do this in a different color we have our D u that's the negative 5 let me do it this way negative 5 cosine of 5x DX negative 5 cosine of 5x DX so that is our D U I'm just changing the order of multiplication times e times e to the U this whole thing now is U this second time around so if we did it this way with just one substitution we could have immediately gotten to the result that we wanted you take the antiderivative of this I'll do it in one color now just cuz I think you get the idea this is equal to negative one-fifth e to the U plus C U is equal to negative five negative sine of five X so this is equal to negative one-fifth e to the negative sine 5 X plus C and you're done so this one is faster it's simpler and over time you might even start being able to do this in your head this top one you still didn't mess up by just setting u equal to sine of 5 X we just had to do an extra substitution in order to work it through all the way and I was able to do this video despite the crowing crow outside or squawking crow