let's integrate a cosine to the fourth now we're going to use one of our power reduction formulas we're going to use cosine squared but right now we've got cosine to the fourth so the very first thing I'm going to do is to change this into a power of 2 all to the second power so we're going to rewrite this as cosine squared 5X all to the second power and then a DX now I can apply my formula for the power reduction of cosine inside of the parenthesis as I do that I end up with the integral parentheses here comes my power reduction formula on the inside I get that fraction 1 plus cosine 2 times Theta but in this case Theta is equal to 5x so that's going to be 2 times 5X um all divided by two and this is all squared and then a d x let's start simplifying this in my numerator I'm going to need to take that binomial 1 plus cosine and this is a 10x I'm going to need to square that binomial in the denominator I'm squaring 2 so that becomes a 4 and this is a DX as I continue to simplify let's bring that 4 out in front and think of that really as a 1 4 and then we can square what I've got on the inside okay so what do I have I've got a 1 4 here I need to square that 1 plus cosine 10x so I'm really going to foil this guy through so I'm going to take that times itself 1 plus cosine 10 X 10 X and then all a DX okay let's go ahead and foil I'm going to need a lot more room and let's keep going okay so doing our foil first is going to be one times one my inner is a 1 times cosine 10x my outer is also a 1 times cosine 10x that's going to be two of them so plus 2 cosine 10 x that's a cosine and then on the end cosine times cosine so I get a cosine squared so cosine squared of 10 x and this is all of course a DX now as I work my way through here I can take the integral of 1. let me actually write down what I've got here so this is a 1 4 times everybody I can take the integral of 1 the integral of 1 DX is equal to X I can also integrate 2 cosine 10x as I take the integral there I get 2 is my multiplier it's positive the antiderivative of cosine is sine 10x but I've got that multiplier 10 on the inside so I end up dividing by a 10 and you could check that with a u substitution and then on the end there I'm not ready to integrate that so I'm going to write this as plus the integral of cosine squared 10x d x that last integral is going to need another Power reduction now there is a plus c floating around I'm just going to hopefully remember to put that in at the very end let's move this up I'm also going to multiply our 1 4 through otherwise I just know I'm going to forget it somewhere along the way so I've got 1 4 times x that's going to be an x divided by 4. let's reduce the two tenths that gives me a 5 in the denominator and then 1 4 times that 5 is going to be a 20 in the denominator and I get the sine of 10x and then I've got plus 1 4 cosine squared of 10x let's go ahead and apply that formula to our cosine squared now here Theta is equal to 10x so when I get to the double the angle formula I'm going to use a 10 X okay so we end up with 1 plus cosine of our angle doubled so that's going to be 10 times 2 so 20x all divided by 2 and there is a DX there and I'm just going to drop my plus C for now I'm going to keep all of this stuff so I've got my X over 4 I've got my sine of 10 x over 20 over 20. so plus let's take that divided by 2 out in front also so I'm going to take that 1 4 and multiply it times one half so that becomes a 1 8 out in front on the inside I've got a 1 to integrate and I've got a cosine of 20 x to integrate DX so this is x divided by 4 plus I still have my sine of 10x divided by 20. plus let's go ahead and integrate this I've got 1 8 and I'm going to take the integral of 1. the integral of 1 is X so I have plus 1 8 times x plus I'm going to do 1 8 times the integral of cosine so plus 1 8 the antiderivative of cosine is sine so I get a sine of 20x and because I've got that multiplier of 20 I'm going to divide by that 20 I'll throw this plus c in there oh my gosh I am just about there and here comes our answer so I've got my X fourth I've got my sign 10x over 20. plus X over eight you could combine these plus sine of a 20x all divided by a hundred and sixty plus a c now if you did combine those those would end up to be 3x divided by 8 but we're going to call it good we've got our answer here you're doing so great take a look at this next video and thanks so much for watching