evaluate if m is natural

[knotty perch]

oh boy

$x^m=k$

[sour dock]

let

[knotty perch]

$\int(\sqrt[m]{(k^{3m}+k^{2m}+k^m)(2k^2+3k+6)})dx$

[sour dock]

let

[knotty perch]

Agh I'm too lazy to multiply that

yeah

hmm how?

mhm

Yes I did

That's why you have $k^{3m}+k^{2m}+k^m$

[sour dock]

let

[knotty perch]

I've defined k as $x^m$

[sour dock]

let

[knotty perch]

Why not?

$k^{3m} = x^{m^{3m}}$

[sour dock]

let

[knotty perch]

So when you get $x^{3m}$ inside the root it becomes $(x^{3m})^m$

[sour dock]

let

[knotty perch]

So that's basically $(x^m)^{3m}$

[sour dock]

let

[knotty perch]

Mhm, yes

So that's just k$^{3m}$

[sour dock]

let

[knotty perch]

So that's what I did here

You simplify this

and then the idea is that you can effectively remove the root

You won't need substitution

One sec, I'll remove the root

Yes, but when we remove the root you'll get x

I'm too lazy to simplify it so let me get wolfram

see, doing that by hand is pain

just...pain

[knotty perch]

Basically, we know k=x^m

so

If you're taking the mth root of this you're effectively removing that exponent

So you can just replace k with x

Mhm

basically, $\sqrt[m]{k^{m+1}}=\sqrt[m]{x^{m(m+1)}}$

[sour dock]

let

[knotty perch]

m and m cancel each other

you get x^m+1

Basically, we have replaced k with x by taking the root

sure

[knotty perch]

Right

Sorry, went offline

Let's do this step-by-step

$3k^{m+1}+2k^{m+2}+3k^{2m+1}+2k^{2m+2}+3k^{3m+1}+2k^{3m+2}+6k^{3m}+6k^{2m}+6k^m$

[sour dock]

let

[knotty perch]

now

let's substitute k with its definition

x^m

$3(x^m)^{m+1}+2(x^m)^{m+2}+3(x^m)^{2m+1}+2(x^m)^{2m+2}+3(x^m)^{3m+1}+2(x^m)^{3m+2}+6(x^m)^{3m}+6(x^m)^{2m}+6(x^m)^m$

[sour dock]

let

[knotty perch]

yeah?

Now, using the laws of exponents, we can multiply the exponents in each term

or

Let's take x^m as the common term

nvm

$3x^{m(m+1)}+2x^{m(m+2)}+3x^{m(2m+1)}+2x^{m(2m+2)}+3x^{m(3m+1)}+2x^{m(3m+2)}+6x^{m(3m)}+6x^{m(2m)}+6x^{m(m)}$

[sour dock]

let

[knotty perch]

Yeah?

we've just multiplied the exponents

Pretty much, yea

Except, I've combined m with the other exponents

basically $(x^k)^m=x^{km}$

[sour dock]

let

[knotty perch]

I've used that here

ok so

[knotty perch]

Let's take the mth root of this

wait..

Ah it's addition

You can't really distribute it like that

yeah, took me a while

I suppose you can sub it with dk

well, if n is natural x^m should be differentiable, right?

Well if you wanna know

[knotty perch]

Oh what's it?

$2x^{2m^2}+3x^{m^2}+6^m$

[sour dock]

let

[knotty perch]

6x^m

The entire expression?

Ah right sorry

$2x^{3m}+3x^{2m}+6x^m$

[sour dock]

let

[knotty perch]

So did you get t?

Ah fair enough

Damn, I can't believe I didn't see that

not seeing these steps is gonna kill my math skills

But yeah, tons of practice is required

$\frac{sin^6x + cos^6x}{sin^2x+cos^2x}$?

[sour dock]

let

[knotty perch]

Ah right

[sour dock]

meckron

[knotty perch]

$(sin^4x+sin^2xcos^2x+cos^4x)(\frac{1}{sin^2x}+\frac{1}{cos^2x})
= (sin^2x+cos^2x+cot^2xsin^2x)+(cos^2x+sin^2x+tan^2xcos^2x)$

[sour dock]

let

[knotty perch]

Without subbing?

oh that's gonna kill me

sorry, never took an integrals course

yet

$\frac{1-3sin^2xcos^2x}{sin^2xcos^2x}=\frac{1}{sin^2xcos^2x}-3\ \int{k} = \int{sec^2xcsc^2x}dx - 3x = tanx+cotx-3x+C$

[sour dock]

let

[knotty perch]

smth like that?

can't get it, brain's dead rn

How do you make the numerator a perfect square there?

right