Integrating 1/(1+cosx) in Two Ways

If you need to post a picture of your solution or idea:
#CalculusProblems #Integrals

EXPLORE 😎:

PLAYLISTS 🎵 :

[MathElite]

Wow an interesting question!
My first move would be to multiply both sides by (1-cosx)

[skwbusaidi]

Multiply top and buttom by 1-cosx
I= INTEGERAL OF (1-cosx)/(1-cos^2x)=integeral of (1-cosx)/sin^2x
=inteferal of csc^2 x- cscx cotx
= -cotx + cscx +c

[aashsyed]

0:52 it's the weierstrass subsitution!

[rohankumarroy]

Given Integral can be written as it is nothing but a simple integral

[thfchris]

My secondary level had not teached the derivative or integration of inverse trigonometry. I would also let t=tan(x/2). Therefore dt=[(1/2)(sec(x/2))^2]dx => 2dt/(1+t^2)=dx... cosx=(1+t^2)/(1-t^2) and so on.

[tonyhaddad]

Its always better to look at problems from different prespective !!!! Nice integrale Good job !!!!

[stewartzayat]

The first method, also known as Weistrass substitution, is definitely worth remembering, since it can be used to reduce any integral of a "rational function in sine and/or cosine" to an integral of a rational function. And those can be algorithmically solved by partial fraction decomposition, for example.

[lukastillman]

a third way would be to to expand the fraction by 1-cos(x). The answer would then be csc(x)-cot(x)+C

[emanuellandeholm]

3rd method: recognize that 1 / (1 + cos x) is a geometric series. Use Euler to rewrite cos x as a sum of complex exponentials raised to integer powers. Use the binomial theorem to expand in terms of e^ikx and e^-ikx. We now have am indefinite integral of a double sum of complex exponentials. Since |cos x| <= 1 we should be OK to switch the order of integration and summation (it really depends on the limits of integration). Obviously I'm BSing and handwaving, but I think this could be a viable and really complicated way to do the integral. :D

[math]

you are excellent with integrals bro, great job solving in two ways

[זאבגלברד]

We have a "formula" that we use: integral of f(mx+b) is (1/m)*F(mx+b) + C . F(t)' is f(t).

[별의별-hb]

I was amazed at using the following formula !!
cos2x = 2 (cosx)^2-1

It would be easy if we had come up with that formula, otherwise it would have been difficult.

[tayelaboulyazeed]

Multiply by the Conj. (1- cos x) both of numen. and denum.

[manojsurya]

The second method was so simple and I like it, the weierstrass sub method was a good but lengthy approach 👍

[ramaprasadghosh]

1+ cos(x) = 2 cos^2( x/2)
so integrant is (1/2)*sec^2( x/2)
Hereby the result is tan(x/2) + const

[kdvyas]

Third method: multiply numerator and denominator by 1- cos x
So, 1- cos x/ sin² x

1/ sin²x - cosx/ sin²x

Then we know the integration of cosec x and -cotx cosecx 👍

[aashsyed]

your channel is growing you are so you are a you are the best

[yoav]

Michael penn solved integral and now you too.great day for integrals:)

[carloshuertas]

Is this a calculus problem? Wow, another great explanation, SyberMath!

[sccheng]

U will get million Subscriber in next next... year (i wait every video from you)