∫√x^6−x^8
#Can you try to solve this problem using trigonometric substitution?
primal knoll
tropic solar
Factor out x^6. That should make it clear.
primal knoll
i did that
still came up with a diff answer from the textbook
can you try solving it for me using trig substitution
tropic solar
sqrt(x^6 - x^8)
= sqrt(x^6 * (1 - x^2))
= x^3 * sqrt(1 - x^2)
x = cos(t)
x^3 * sqrt(1 - x^2) = cos^3(t) * sin(t)```I'm sure you can take it from there.
tropic solar
I got sin^3 (x) times cos^2 (x)
You used x = sin(t), which isn't wrong, but you forgot the substitution was inside a square root.
primal knoll
no
Since dx = cos(x)dx
so I have
x^3 sqrt (1-x^2) * dx
so it becomes sin^3 (x) * cos(x) * cos(x)
no?
tropic solar
Oh, right. I forgot to replace dx. In my defense, you didn't write it.
primal knoll
haha lol
When I fully expand it I end up with a weird answer very close to the textbook
but I cant tell where I deviated
from the correct answer
tropic solar
Okay, so we have int sin^3(t)cos^2(t) dt.
primal knoll
Yes
I then broke down sin^3(t) into sin^2(t)sin(t)
I converted sin^2(t) into (1-cos^2(t))
so I end up with
int (1-cos^2(t))*sin(t)*cos^2(t) dt
cos(t) = u
-sin(t) dt = du
dt = du/-sin(t)
-1 int (1-u^2)u^2 du
taking the antiderivative I get
-cos^3(t)/3 + cos^5(t)/5 + C
tropic solar
Gotta roll it back to terms of x.
primal knoll
plugging in the value of cos(t) through a triangle I end up with
-(1-x^2)^3/2 (1-x^2)^5/2 + C
-------------- + --------------
3x^3/2 5x^5/2
the textbook answer is
(-1+x^2)(2+3x^2)(sqrt x^6 - x^8)
---------------------------------------- + C
15x^3
tropic solar
Okay, wait, I can't read any of that. What's cos(t)?
primal knoll
(1-x^2)
tropic solar
...not according to my math.
Okay, wait, where did the xs in the denominator come from?
primal knoll
when I plug in the values in the right triangle, Cos (t) = adj/hyp so it is sqrt(1-x^2) / x
tropic solar
Why is your hypotenuse x?
primal knoll
because x = sin(t)
tropic solar
And sin(t) equals...?
tropic solar
You didn't even do the wrong math right. If your hypotenuse was x, then your adjacent side should've been sqrt(x^2 - 1).
primal knoll
ok
yea basically
I drew the triangle right and labeled
just wrote down the wrong thing
no the adj side should be sqrt (1-x^2)
tropic solar
...if x was the hypotenuse.
tropic solar
Hence "you didn't even do the wrong math right".
primal knoll
so my final answer is now
-(1-x^2)^3/2 (1-x^2)^5/2 + C
-------------- + --------------
3 5
tropic solar
Where the hell does the textbook get the xs in the denominator?
tropic solar
I am almost entirely certain the textbook is just wrong. I could be completely certain if I felt like differentiating that ridiculous monster.
primal knoll
Maybe its the same answer
just differently written
but this textbooks been giving me so many problems
tropic solar
It's not the same answer.
primal knoll
I've done so many questions
and came up with diff answers
and im certain that all the steps are took are legal
thats why I wanted to talk to someone on a call so we can try to figure it out and think of a new way of solving
tropic solar
,w d/dx (x^2 - 1)(2 + 3x^2)(sqrt(x^6 - x^8))/(15x^3)
wise fjordBOT
tropic solar
Huh. Sonofabitch.
primal knoll
Hahaha
Perhaps they used U substitution however the book requires us to use trig substitution
,w d/dx ((1-x^2)^3/2)/3 + ((1-x^2)^5/2)/5
wise fjordBOT
tropic solar
,w d/dx (1 - x^2)^(5/2)/5 - (1 - x^2)^(3/2)/3
primal knoll
it input the question wrong whatever
wise fjordBOT
tropic solar
Which is also correct.
tropic solar
Or off by a constant, at least.
primal knoll
Hmm
tropic solar
Obvious first step is to combine the fractions. As in, so obvious I don't know why I'm the one who has to say it.
tropic solar
...multiply by x^3/x^3.
tropic solar
You did.
primal knoll
Hes the one who made my brain
im just messing around ik what you mean
thank you though
tropic solar
No prob.
I really have no clue what the hell's going on with the form of that answer.
primal knoll
All the questions and answers in the book is like that. The textbook has videos that answers each problem but the professor refuses to give us access to it and i tried finding them online
I emailed her and fought about it but she doesnt care
tropic solar
,w int sqrt(x^6 - x^8) dx
wise fjordBOT
tropic solar
Okay, what the hell, WA?
primal knoll
wow
tropic solar
Okay, WA did u sub.
primal knoll
thats cheating