Antiderivatives Visually via Accumulation

In this installment of synthwave and math, we sketch the net accumulation curves of elementary functions by measuring the directed, signed area between the curve and the x-axis. This means that area above the x-axis, moving from left to right, is positive. If you move from right to left it is negative, and then these switch below the x-axis. We use these accumulation functions to state general antiderivative rules. Along with various rules (like substitution and integration by parts, etc.) and chain rules, these rules help you find a variety of antiderivatives. If you want proof that these are the correct derivatives, you should consult any standard calculus text.
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For a related video with derivatives, check out:
• Elementary Derivative ...
For more math and synthwave, check out:
• Synthwave Mathematics
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[@Anonymous-sp9zv]

What I failed to learn in high school. You made it totally clear under 5 mins. Thanks very much

[@MathVisualProofs]

Happy to help!

[@Ninja20704]

I am a bit glad with the answer you wrote for the 1/x integral.
Some math teachers and professors have actually said that the usual answer that we give of ln|x| + c isn’t the full answer. Because the domain of 1/x is two disjoint intervals, you can actually have one constant for the positive side and a different constant for the negative side. And it will still be a valid antiderivative.

[@MathVisualProofs]

I was gonna do the same for sec^2(x) but there were just too many intervals… :)

[@soworrounderfalckang1165]

My mind blown up!!

[@Oxygenationatom]

The music goes crazy

[@Mediterranean81]

4:00 why is the area e^x - 1

[@childpr.69]

on the basis of sign so ig they haven't taken mod

[@mynamesgus4295]

-1 is just some arbitrary constant it could have been any real number. He just chose -1 as an example

[@MathVisualProofs]

When you compute the net accumulation starting at 0, you have to have an area of 0 when x=0. But, it turns out that this particular accumulation has to be e^x-1 so that the directed signed area is 0 when x=0.

[@Mediterranean81]

Good

[@MathVisualProofs]

:)