Solving x'=sinx: Where do the x0's Come From?

[evolution685]

i'm having trouble with this.

i first need to solve x'=sinx. the answer is given as t=ln|(csc(x0)+cot(x0))/(csc(x)+cot(x))|]

i'm not sure where those x0's came in from.

here's what i did:

Int(1/sinx)dx=Int(1)dt
Int(csc(x))dx=Int(1)dt
Int((-1)[(-1)csc(x)(cot(x)+csc(x))]/(cot(x)+csc(x)))dx=Int(1)dt (this is the same as the line above but then let's you change Int(f'(x)/f(x)) to ln(f(x)))
-ln(cot(x)+csc(x))=t

so as you can see this answer isn't the same as the answer that was provided. what am i doing wrong and where do those x0's come in from?

any help is appreciated.

 

[HallsofIvy]

What happened to your constant of integration?

If \int f(x)dx= \int g(x)dx then the most you can say is that f(x)= g(x)+ C where C is some constant- and if you are calling your function "x" then "x₀" is a good a name as "C".

Your last line should be -ln|cos(x)+ csc(x)|+ C= t. Of course, - ln|cos(x)+ csc(x)|= ln|1/(cos(x)+ csc(x))| and, of course, - ln|cos(x)+ csc(x)|+ C= ln|1/(cos(x)+ csc(x))|+ C= ln|D/(cos(x)+ csc(x)| where D= e^C is just another constant. The numerator above, csc(x0)+cot(x0), is just a number like "D". I presume they have some reason for writing it that way but it is exactly the same thing.

 

[evolution685]

ahhhhh. i see. thank you.

 

[evolution685]

my next problem is that it says show that for x0=pi/4 you can solve x=2arctan((e^t)/(1+sqrt(2))

here's what i have so far, working from the previous equation:

t=ln|((csc(pi/4)+cot(pi/4))/(csc(x)+cot(x))|
t=ln|((2/sqrt(2)+1)/(csc(x)+cot(x))|
e^t=(2/sqrt(2)+1)/(csc(x)+cot(x))
csc(x)+cot(x)=(2/sqrt(2)+1)/e^t

then I'm stuck. i don't see how to isolate x and where an arctan would come from.

thank you in advance.