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okay so I was making my video and I just got email from one of my subscribers and he asked me how to integrate this and at the bonus I also put this down on the side and thats you know take some time you know pause the video and try this too first okay which one do you guys think is easier hmm in fact they are equally just as easy all right so lets do this one first because this is the question that he emailed me and actually its really easy its conceptional rather than computational okay so first of all the question was given this right here is pi and this is negative pi so of course you know this is bigger than that which is no good and if you allow you can rewrite it this is going to be negative and I change the water front equation and you get negative pipe down here and then pi

that this is just one of the integral property and then you still have the inside two states thats a sine x over one plus x squared plus X to a fourth power DX and now you might be wondering how in the world can we actually integrate this well we dont need to integrate this because first of all I dont think we can actually find answer for this and second of all notice that that power right here on the bottom hey yeah oh if I write this is extra stress power technically and times an odd function oh no this is an odd function for the integral right here inside so I will tell you guys that lets say if you see f of X if you call that to be sine x over 1 plus x squared plus X to the fourth power and to show this is even more odd what you do is you put down F

and the inside is going to be negative x you check right this is the work that you do to convince people they actually have to even what not function so F of negative x this means you plug in negative x into all the X here so you have sine of negative x here and then on the denominator you have 1 plus parentheses negative x and then you square that and then you other words negative x and then you raise that to the fourth power okay on the top son its an odd function so that means negative x you can put a negative to the front this becomes negative and then sign of just X and then on the denominator here this is 1 plus negative 1 squared is just 1 and X square still of course that queer right so this is x squared and then same thing because its even power so in the end you see

that we can actually have a negative right here and then the red part still the original function so this right here we have F of negative x being negative f of X this means this function is aa right this right here is an odd function so when you integrate an odd function and if this is your same number parts like this is negative version of that in the end you get a nice number and then Im number is just 0 and technically you put on negative 0 power Hooters negative 0 right so the answer is 0 so perhaps Ill also write that one note on the side this right here it says if you have negative a to a of an odd function so to put an odd function like this the Y over the CTX this right here is just equal to zero this is one of the integral properties now as a bonus how would you

this without any number here this is just not patent either because we are integrating with respect to Y why am I doing this huh talk to the street kind of so in the wide world this is just like accounts then that means all we have to do is just write this down and put a white next to it so its a constant multiple is put on sine of X over 1 plus x squared plus X to the fourth power and then multiply by Y but not technically put plus C and thats it and why dont we put on plus C you just let me know by the way thats just for fun by the way this is how you integrate even though you have a crazy integrand but if the insight is an odd function we have this property and thats a main point of this little scene right here as always thats it thats just for fun right

blackpenredpen, math for fun, bprpLIVE
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