integral of 1/(sqrt(x)-cbrt(x))

Integral of 1/(sqrt(x)+cbrt(x)), integral of 1/(sqrt(x)+cube root of x)
what should we choose to be the u?

Integration with partial fraction, calculus 2 tutorial
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#blackpenredpen #math #calculus #apcalculus

[mrmeowtv]

At first I thought, hmm is it x^(1/6)? Then I thought nahhh that would complicate it too much. Then I watched the video. Oof

[sardarbekomurbekov]

I liked the beautiful answer when coefficients of each term equals to degrees of the roots

[taranmellacheruvu]

4:00 You could alternatively add one to the numerator and then subtract one from it, so that you can factor and cancel out terms.

u^3 / (u - 1)
(u^3 - 1 + 1) / (u - 1)
(u^3 - 1) / (u - 1) + 1 / (u - 1)
(u - 1)(u^2 + u + 1) / (u - 1) + 1 / (u - 1)
(u^2 + u + 1) + 1 / (u - 1)

Thank you, blackpenredpen, for teaching this useful technique in your other videos. It has helped me a lot recently.

[krrishmaheshwari]

I USED A³-B³ Identity instead of long division!

[MG-hish]

Incredible. It would've taken me ages to see that. What a method. What a guy.

[-rahul-]

synthetic division saves a lot of time 😅

[balajishankar]

You are a genius, and a great teacher. Love your videos. If I may ask, where did you do your Masters in Math?

[rubetz]

Hey Steve!

Great to see you making calculus look cool to the world! Do you think your could do a more detailed video on dividing polynomials? That's one very elegant move to me. Cheers!

[pierreabbat]

"... and u will C y." u is the sixth root of x, and C is the constant of integration, but what's y?

[holyshit]

Will you record video on integration of binomial differentials
Integrals of the form Int(x^m(ax^n+b)^p dx)
There are substitutions which rationalizes integrand of this integral
This is the first case of the integral above
4:19 if we dont want long division we can use difference of cubes
(We call it short multilication formulas)

[manishkumarsingh]

I used the same technique u show how to rationalize 1÷ {3^(1/3)-√2}
Thx to u😊😃

[muratkaradag]

Amazing. Im writing it down with my blue pen.

[sanachanjjangg]

Thank you for solving a problem I was still stuck with at 3am :))

[braydenhaines]

Does cosec^-1(x) equal sin(x)? Because cosec(x) equals 1/sin(x), would the inverse of that be just sin(x)?

I need this for my maths investigation so I can revolve a solid around the y-axis, I just need the function of y (like x= 2y^2) of y=2cosec(x+Pi/2)-2. I have converted that into something but I need to know what cosec^-1(x) is first or if I’ll have to cross multiply. Any help is greatly appreciated!

[adude]

Can't believe I did this. Granted, I used two substitutions (u = cbrt(x) and v = sqrt(u) - 1), and at the end I had a -6 mixed up in the result, which is constant so I could just add it into the C

[silasrodrigues]

About the simplification of u^2 cancelling in numerator and denominator should take us to analyze the domain of the result since u could not be zero. But in the domain of original function x¥0 and x¥1 and as long as 'u' is related to x^(1/6) the restriction is automaticly applied and we don't need to worry about the division by u^2. (I used ¥ meaning 'different').
Am I right?
Sorry if it's messy, my english is not that good...

[ericherde]

I was off by a couple minus signs because I solved the integral in the title instead of the integral in the thumbnail. :Þ

[draaagoo]

i did it by two substitutions and then i addec1 and substarcted it

[UnathiGX]

how do u do the u substitution instead of the long division?

[MrRyanroberson]

I wonder if there's some trig type function where f^3 + g^3 = 1, for g(x) = f(pi/2-x), or with something other than pi/2... Anyways, seeing those exponents made me think to use x^(1/6), and end up with 1/(x^3-x^2) and partial fraction for A/x^2 + B/(x+1)