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What substitution would i use for
the integral of (Sec(x)^9)(tan(x)^3)
>>8095855
ah thanks
>>8095853
cos(x) = u
-sin(x)dx = du
sin^2(x) = 1 - cos^2(x)
>>8095853
Don't use substitution
Use trig identities, break it down as far as you can then work it as a power function
>>8095853
let t=tan(x/2)
then cos(x) = (1-t^2)/(1+t^2)
sin(x) = (2t)/(1+t^2)
tan(x) = (2t)/(1-t^2)
this allows you to turn any trigonometric rational function into a rational fraction that's trivial to integrate.
>>8095871
I'm so mad I didn't learn that before diffeq. Cheers from someone who needs to get back to math.
>>8095853
u=log x
thank me later
>>8095871
Wat how do you do this
>>8096169
do what? change of variable? or integrate?
>>8096174
I dont see why
(1- tan^2(x/2))/(1+tan^2(x/2)) = cosx
>>8096169
tan(x/2) = u
sec^2(x/2)*(1/2)dx = du
so, if your integral is in the form int[ R(sin(x),cos(x))]dx, then you can change it to
int[R(sin(x),cos(x)]*2cos^2(x/2)*sec^2(x/2)*(1/2)dx, which becomes int[R(((1-u^2)/(1+u^2)),(2u/(1+u^2)))]*2*(sqrt(1/(1+u^2)))du
>>8096176
[math]\frac{ 1 - \frac{sin^2(x/2) }{ cos^2(x/2) } }{1 + \frac{sin^2(x/2) }{ cos^2(x/2) } }[/math] is the same as
[math]\frac{ \frac{cos^2(x/2) - sin^2(x/2) }{ cos^2(x/2) } }{ \frac{cos^2(x/2) + sin^2(x/2) }{ cos^2(x/2) } }[/math]
if you simplify, you get [math] cos^2(x/2) - sin^2(x/2) [/math] which is [math] cos(x)[/math]
>>8096182
i fucked up the 2cos^2(x/2) calculation, the sqrt shouldn't be there/
Why not just set up the integral like
Sec(x)tan(x)sec^8(x)(sec^2(x)-1)
And then sub u=tanx. Come on /sci/ all these complicated methods when all u have to do is play around with trig identities and simple u substituion.
>>8095871
https://www.researchgate.net/publication/281255489_Rotor_Coordinates_Vector_Trigonometry_and_Kepler-Newton_Orbits
In case you want an intuition for what this "t" is.
(It's the half-turn in the paper.)
>>8095853
All trigonometric integrals goes like this:
2cos(x)=e^ix+e^-ix
2isin(x)=e^ix-e^-ix
>>8096225
It's from the rational parameterization of the unit circle. Do you even rational trigonometry. The t gives a point (0,t) on the y axis and the coordinate (cos(x),sin(x)) is where a line from (-1,0) through (0,t) intersects the unit circle. t=tan(x/2) is due to the inscribed angle theorem.
>>8095869
>i make things as difficult as possible so i can brag about it on 4chan
This is all obvious for me, but how do I prove such statements?
>>8096773
oh shit, wrong thread
>>8096773
hint: norm.
