Integral battle#18, u sub (GONE LONG)

integral of x^(1/2)/(1+x^(1/3)) vs, integral of 1/(x^(1/2)-x^(1/3),
integral battle

blackpenredpen

tags: integral of x^(1/2)/(1+x^(1/3)), integral of 1/(x^(1/2)-x^(1/3), integral of x^3/(x-1), integral calculus, integral examples, hard integral problems, integral with substitution, integral with long division

[Kanbei]

Instead of long division for the second one you can do -1+1 in the numerator to give ((u^3-1)+1)/(u-1).

This means you get the difference of cubes in the numerator x^3-1^3 = (x-1)(x^2+x+1)

the (x-1) cancels leaving x^2+x+1 + 1/(x-1) to be integrated

[Liooooooooooooooo]

Is it just me, when I see how he solves these complex integrals I feel so satisfied and relaxed in a weird way, just me? OK

[sribasnath]

You can't imagine how much you helped me

[SuperDreamliner]

Hey :)
Small mistake with the second integral, for the u sub you wrote du= 6u^5 du instead of dx=6u^5 du
Keep up the great work :)
Greetings from Germany :)

[joshuasorber]

You should make a video that goes through the thought process for deciding which method or determining harder U-Sub problems and things to look for etc. That's the biggest problem I and a lot of people struggle with. Thanks!

[thundercraft]

It's my first time trying some of the integrals in your vid before watching it
And i got the two right

[hollynessgoldwater]

thanx moreee please can u do this question cos^3x/sin^2x+1

[sardarbekomurbekov]

Great you helped me to solve similar integral. Thank you!

[Humongastone]

when you took the 6th root on both sides at around 7:08, do u need to put the "+/-" ?

[kylegatesman]

Without using the hypergeometric function, what is the integral of 1/(1+x^n), for any positive integer n? I have seen improper integrals of this before, but an indefinite integral of this would be nice to see.

[craigs]

I totally forgot about algebraic division! I would have wrongly tried partial practions. Good video

[CTJ]

awesome vid - great explanation - thanks

[orelfrank]

um isnt sqrt of something to an even power absolute value ?

[richardfredlund]

I pause and try... then blackpenredpen makes it look so easy... and I'm thinking why couldn't I do that. I tried u sub x^1/3 and got stuck.

[aeroflySL_]

bprp using trig sub x = tan^6tita i solved a bit but ending up with integral of tan^8tita, , then again its hard long right any shortcut?

[deeptochatterjee]

I think it's interesting how the final answer if the second one has so many normal looking anti-derivatives from the power rule. Was there some lurking long division there in the x-world?

[sam-kxty]

This is the same thing as rational substitution right ?

[deeptochatterjee]

10:21 Couldn't you have subtracted 1 from the numerator and then added the fraction 1/u-1? You then factor the first fraction to be u^2+u+1 and then the second fraction integrates to ln|u-1| to get the exact same answer.

[lakshaymd]

That title....I see what you did there 😂😂😂

[helloitsme]

It's such a shame we in Holland never learned long division or I forgot it lol