Integral of x^i

Learn how the complex integral of x^i can help us find the integral of sin(ln(x)) and cos(ln(x)) without using integration by parts. Complexifying the integral is an integration technique that is usually not taught in calculus 2.

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#blackpenredpen #math #calculus #apcalculus

[gcewing]

"Killing two birds with one stone" is a strangely appropriate metaphor, seeing as "calculus" is the Latin word for "stone". :-)

[OLApplin]

6:55 "I don't like to be on the bottom, I shall be on the top "... your choice man!

[lukevaneyk]

I feel that teaching with colors as you do aids a lot in the grasping of these problems. When reading textbooks, it isn't always clear how they get from one step to the next. Your demonstrations make it much clearer.

[suave]

This channel is awesome *isn't it?*

[CODPIMPS]

I love your channel. I'm currently in high school, working on my second year of calculus, and I love challenging myself with your higher level content, and your explanations are easy to follow and in depth enough, but not too in depth to the point of being boring. Keep up the great videos.

[adb]

The title of this video should not be "integral of x^i" but "how to solve integral of sin(ln x) and of cos (ln x) at the same time and much easier than using substitution and integral by parts"

[verainsardana]

10:35 "if you are obsessed with +c" i laughed so hard

[mikety]

One last thing though - technically, you would need to prove the power rule in fact is valid for complex numbers to make this airtight, though I suppose you're going on that that's already been done: nonetheless, it wouldn't necessarily be justifiable at, say, the usual Calc II-like level where this seems to target, to do that.

But that is easily remedied: you can first assume _as a heuristic or hypothesis_ that the power rule will work for n = i (seems reasonable, no?) and then after carrying through with it, go back and _differentiate your final answer_ to see that you do indeed get cos(ln(x)) + i sin(ln(x)), thereby proving that not only does the rule apply for that imaginary power but also that you have indeed integrated the two integrals that you wanted to "with one stone". This is not circular because you did not reference the conclusion as justification; rather simply only took it as a hypothesis to then later be verified.

[whozz]

I recommend you to see the graph of the integral of sin(ln(x)). It is really cool, the graph repeats itself in ever larger scales as x grows to infinite.

[AndDiracisHisProphet]

and some people say imaginary numbers aren't useful

[АлександрЮсько-уд]

I love how happy you got after seeing this patern. I've recently stopped focusing on mathematics, because of my software engineer carrier, but that video reminded me the way I felt after solving things. Thank you for the video.

[toastybread]

Can we take a second to appreciate the pen lid removal at 6:16

[evilotis]

i love how this dude sits and works out crazy-ass maths problems on his own at home, and when he comes up w something great, he shares it w everyone on YouTube. his excitement is infectious, and i love watching his channel <3

[LeeSawyer]

I've become addicted to this guy's videos!

[premdeepkhatri]

Best Solution of the problem. Very deep thinking require for this video to create. Thank you Blackpenredpen.

[disasterarea]

this is the most wholesome channel on youtube

[alexturk]

love how he switches so seamlessly between marker colors

[Jelimox]

at the end, you missed the "i" at the final result

I'm more excited about your excitement than the solution itself. :D But I understand the genuise way (I could never figure out by myself). Really good job! Booth thumbs up.

[strafeae]

One of my favorite things about this channel is that you respond to EVERY SINGLE COMMENT. You really do make YouTube a greater place not only because of your mathematical genius, but because you are just a wonderful person <3