Question

Using the substitution x = cos^2(x) OR OTHERWISE, find:

Answer 1

\[\int\limits_{}^{}\frac{ 2+x }{ \sqrt{x^3(1-x)} }dx\]

Answer 2

I've substituted it, but am stuck.

Answer 3

after substitution did you get
\(\huge \int -2\dfrac{2+\cos^2 x}{\cos^2x}dx\)
??

Answer 4

no, but wait, lemme change a few things. I turned 1-cos^2(x) into sin^2(x), should I have done that?

Answer 5

YES.

Answer 6

then ?

Answer 7

and the derivative of cos^2(x) can be written as -sin(2x)??

Answer 8

no.....

Answer 9

i mean, sorry, but it was cos^2(theta), not x, so we would need to change the dx to dtheta.

Answer 10

no??

Answer 11

derivative of cos^2(theta) can be written as 2 cos theta * d/dtheta (cos theta) =... ?
chain rule

Answer 12

which is -2sin(theta)cos(theta).

Answer 13

yes

Answer 14

but 2sinxcosx is sin2x, so why can't we make it negative?

Answer 15

but you got how ?

Answer 16

i got what how?

Answer 17

oh...ok...its corrct, -2sin 2x but don't write it that way
keep it as -2 sin x cos x

Answer 18

ok, but it should be just-sin(2x), not -2sin(2x), then, i guess :P

Answer 19

yeah :P

Answer 20

ok, got that.

Answer 21

then what gets cancelled ?

Answer 22

I'm guessing the root, but idk how.

Answer 23

\(\huge \sqrt {\cos^6x \sin^2 x}= \cos^3x \sin x \)

Answer 24

OMG HOW DID I NOT SEE THAT. KILL ME. NOT LITERALLY.

Answer 25

then one sin and one cos from both numerator and denominator will get cancelled and you will get
\(\huge \int -2\dfrac{2+\cos^2 x}{\cos^2x}dx\)

Answer 26

Ok, I have that. now wait, lemme see what I can do.

Answer 27

now its a piece of cake :D :P

Answer 28

split the fractions, ugh.

Answer 29

yes.

Answer 30

hold on, if x = cos^2(theta), then can we say that (x)^0.5 = cos(theta), and therefore theta = cos'(x^0.5) ??

Answer 31

yes, but what is that needed ?

Answer 32

because we have 1, and when we integrate that we get theta, and we need to give the ans in terms of x :P

Answer 33

ohhhhh.....yes.
but do you have an answer to verify ?

Answer 34

nope, sorry, this kinda worksheet is like a take home test. No answers until he corrects it and I get everything wrong :P lol jk.

Answer 35

haha..ok
\(\large \cos^{-1}\sqrt x\)
in the answer is pretty weird but its correct here...

Answer 36

and so is \(\large \tan [\cos^{-1}\sqrt x]\)

Answer 37

ok THANK YOU SO MUCH

Answer 38

ok WELCOME SO MUCH ^_^

Answer 39

and you better hang around, I might post a few more q's.

Answer 40

sorry :(

Answer 41

sorry??

Answer 42

i was leaving....