integral of abs(x)

Integral of abs(x),

blackpenredpen

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Thank you,
blackpenredpen

[michaelwood]

Thank You. I'm a 60 year old Engineer and probably will never have a use for most of the calculus I have learned (or to be learned).... But I wanted you you know I religiously watch your videos and enjoy ALL OF THEM !!! Keep up the great work!!!

[RaSkipper]

Great video, but why were you holding a thermal detonator the whole time?

[mikety]

This answer can be written more compactly using the sign function:

int |x| dx = 1/2 sgn(x) x^2 + C.

[mokoufujiwara]

If you want a solution in ONE line, you can write (x*|x| / 2) + C, or sgn(x)*x^2 /2 + C, period.

[jamesnguyen]

Me: Alright, let's go do my calculus homework.
Also me: Oooo I haven't seen this blackpenredpen video yet.

[magnuskonig]

"Enough talking!“

proceeds to talk

[Cannongabang]

could've done 1/2 *x*abs (x) +c as integral. thought it was a good answer ahha

[john-]

I prefer to express the antiderivative as |x|/x * x^2/2. |x|/x gives 1 if x is positive and -1 if x is negative

[mahmoodayesh]

int( |x| ) = int( |x|/x * x) = |x|/x int(x) =
|x|/x * x^2/2 =1/2 * |x| * x
You can actually do that because the derivative of |x|/x is 0 so it is technically a constant...

[tejloro]

Could you write the final answer as (|x|/x)*(1/2)(x^2)+C since |x|/x = -1 when x<0 and +1 when x>0 (Yeah, i know there's an issue when x=0, but if that's not the case it should work, right?

[TheDigiWorld]

1:04 this is not "lol" 😂😂😂 really love your sense of humor

[cipherxen]

Watched a video on another chanel where they use abs (x)=sqrt (x^2). And answer was 1/2*x*abs (x)+c.

[EpicMathTime]

Man I just found this, glad we did things a bit differently. 😂

[nebrosarth]

You should also put tanΘ into abs after removing square root and power.

[Hacker]

but... |x|² is still x², if integral of |x| is -½x² that would be a negative area, but in face that is a positive area in this case?

[JSSTyger]

The value of this channel is "absolute"...BADDUM

[blockthrower]

in the end you can also compress the piecewise function of the integral to a single function: 0.5x*|x|

[laxmikandari]

Sir so much thanks to you i am doing coaching in india for iit and i was stuck at it ....our teacher has n't reach to modulus so i cant solve this... but thanks to you sir ... really ..love from india❤

[kepler]

This would’ve been so much easier if there were values for the integral

[douglasshamlinjr.]

I ended up with 1/2*x*|x|.

Derivative of |x| = |x|/x

derivative of 1/2*x*|x|
= 1/2*(x*|x|/x + |x|*1) (Product rule)
= 1/2*(|x| + |x|) = |x|

A trick about |x| in integrals is to multiply by x/x to obtain x*|x|/x. Then treat |x|/x as a constant (it is either 1 or -1) and bring it out of the integral then integrate what's left.