Integral of absolute value of x (abs x)

In this video we're finding the Integral of absolute value of x (abs x). Enjoy!

#math #brithemathguy #integral

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

[BriTheMathGuy]

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[EpicMathTime]

True warriors use integration by parts.

[jasonmcclatchie]

I never get tired of the way you manage to calmly simplify and explain these problems. Keep up the great work.

[adrianschmidt]

In the end it would be nice to write it as 0.5x * |x| +C

[MichaelRothwell]

Since |x| is piecewise of the form ax^n (where "a" varies between pieces but n is fixed), you can use the power rule in the form: to differentiate, multiply by the power (1 here) and divide by x; to integrate, multiply by x and divide by the new power (2 here). Hence the derivative of |x| is |x|/x, and its antiderivative is is x|x|/2 + c.

[anshumanagrawal]

Just differentiate abs(x) by writing it as sqrt(x^2), and then using integration by parts and the derivative of abs(x) that we just found, you can see that the integral of abs(x) is [ x abs(x) ]/2 +c

[smartphonesammler]

I found another way to solve this integral. I multiplied the integrand with one and then used integration by parts. ∫(1*|x|)dx = x|x| - ∫(x*|x|/x)dx I used the fact that the derivative of |x| is |x|/x and then the two x cancel out and you have: ∫|x|dx = x*|x| - ∫|x|dx Now we can bring the second integral to the other side: 2*∫|x|dx = x*|x| + c And then we divide by 2: ∫|x|dx = (1/2)*x*|x| + c.

[kilian]

You can also express the antiderivative as |x|x/2. Also, the derivative of |x| can be expressed as |x|/x.

[jakubzagrodzki]

you can write the solution as (abs(x)*x)/2

[eliyasne]

I find it much more useful to generalize to the folowing:
For all f(x) = |x|·x^n
f’(x) = (n+1)·|x|·x^(n-1)

[theimmux]

2:00 Integrating piece-wise functions you should use two distinct constants instead of just C. Maybe C and D or just C_1 and C_2. Nothing says that they should always have the same value

[angelmendez-rivera]

I would use integration by parts. By realizing that d(|x|)/dx = x/|x|, you can say |x|·dx = d(x·|x|) – x·x/|x|·dx = d(x·|x|) – x·|x|/x·dx = d(x·|x|) – |x|·dx. Therefore, 2·|x|·dx = d(x·|x|), so |x|·dx = d(x·|x|/2). So the antiderivatives of |x| are equal to x·|x|/2 + C, and this is equivalent to what you have in the video.

d(|x|)/dx = x/|x| is obtained by considering |x| = sqrt(x^2) and using the chain rule.

[OptimusPhillip]

And for a definite integral, just sum the definite integrals of the two pieces

[TheDigiWorld]

I just love watching your videos. And funny enough, a thought came to my mind out of nowhere: if we have to find the integral of |x| from a to b, we might be able to find it without calculus, just applying some basic geometry using the graph of |x|. But what if there are no bounds? And by some weird coincidence I just happened to come across this video

[bradensorensen]

Would this not be x squared?
Missed the divide by 2 part in my hasty in my head method.

[RedTitan]

... Thank you for explaining it in detail sir

[coco]

This was a cool video, thanks💯 also pls make a video on how to solve difficult integrals .

[砖递鷵橡孂㨩錘墭]

what about the intergral of the absolute value of a polynomial

[helloworld-hi]

Also known as |x^2/2|
Edit: MY SCREEN FLIPPED OUT.💀

[mohammednasser]

What if you wrote it as x|x|/2 + c
Instead of that ?