Integral of absolute value of x or abs(x)

This calculus video tutorial explains how to find the integral of absolute value of x or abs(x) using graphs and piecewise functions as well as the power rule for integration.

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[tatyana]

This man has a video for everything I search man. Love him

[georgesadler]

Professor Organic Chemistry Tutor, thank you for an awesome video on the Integral of the absolute value of x. This is an error free video/lecture on YouTube TV with the Organic Chemistry Tutor.

[umergulmuhammad]

More examples of integration with these | |
Example:
|ax+b|
|ax2+bx+c|
|sin(×)|
|cos(x)+sin(x)| More trigonometry examples

[thegddevil]

It's also 1/2*x*|x|+C
See more has proof





x|x|
if we take the derivative, we get:
1*|x| + x * d/dx (|x|) via the product rule
*derivative of |x| is sign(x)*
simplifying, we get:
|x|+xsign(x)
x, sign (x), and abs(x) all are related
x = abs(x)/sign(x) *x is not 0, then you get 0/0*
sign(x)=abs(x)/x or x/abs(x) *x is also not zero, then you also get 0/0*
and abs(x)=x * sign(x), or x/sign(x) *for x/sign(x) x is not zero, as you, once again get 0/0*
abs(x) = xsign(x)
xsign(x) is in the derivative, so lets substitute x sign (x) for abs(x)
|x| + |x|
2|x|
2|x| is the derivative of x * |x|
Now we have |x| in a derivative, so we can isolate it.
d/dx (x|x|) = 2|x|
divide both sides by two
1/2 (d/dx (x|x|)) = |x|

*when you have d/dx ((constant) * f(x)), you can factor out the constant, so now lets factor IN the constant,
d/dx (1/2*x*|x|) = |x|
The antiderivative of |x| is x|x|/2 +C

qed

I don't know if this has been proven somewhere else, im in 10th grade so i havent had a lot of time to search, this video came up when I googled it and nothing else like the proof in this comment have I seen anywhere else.

thank you for reading

[ishangupta]

Pls upload video of reaction with hio4 (periodic acid)

[chessnoob]

Hey how can it be converted to x|x|/2 ?

[sergeiivanov]

Or: integral of abs(x) is sgn(x)x^2/2 + C

[Geozeff]

Could this also be written as |(1/2)x^2 + C|

[pixel]

int(|x|dx)
u=abs(x)
u'=|x|/x (bprp has a video on it)
v'=1
v=x
=> int(|x|dx)=|x|*x - int(|x|/x * xdx)
<=> int(|x|dx)=x|x|- int(|x|dx)
<=> 2int(|x|dx)=x|x|
<=> int(|x|dx)=x|x|/2 + C

[einsteingonzalez]

What if I don’t treat |x| as a piecewise function?