## Integration of log(cos(x))?

Math help on Calculating limits of functions, Differentiation of a product, Differentiation of a quotient, Composition of functions, Chain Rule, Variations, Extremums: Maximums, Minimums, Increasing Functions, Decreasing Functions, Constant Functions, Differential Equations, Growth of a Population, Formulas, Integration by Parts, Change of Variables; Double Integrals, Triple Integrals, Multiple Integrals, Integrals with a Parameter, Area, Volume, Approximations on My Math Forum.

### Re: Integration of log(cos(x))?

ZardoZ wrote:With complex analysis we may follow the same steps presented here.

Yes, I know. But you can't use complex analysis to find an elementary anti-derivative of log(cos(x))! (As OP's claim )
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3758
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### Re: Integration of log(cos(x))?

mathbalarka wrote:
ZardoZ wrote:With complex analysis we may follow the same steps presented here.

Yes, I know. But you can't use complex analysis to find an elementary anti-derivative of log(cos(x))! (As OP's claim )

The only relevant post to what you are saying is this one

It is possible to exactly calculate the definite integral of this if you go from one multiple of pi/2 (including zero) to another multiple of pi/2 and requires no special functions or complex numbers.

made by "The Fool" (who in reality is the opposite ), who clearly states that he is talking about definite integration.

ZardoZ
Super User

Posts: 1664
Joined: Mon Nov 08, 2010 5:17 am
Location: Greece, Thessaloniki

### Re: Integration of log(cos(x))?

Yes, but OP says (s)he want's the calculation of the indefinite integral in terms of elementary functions:

Najam wrote:We want to calculate it by using indefinite integration.

Najam wrote:I am interested in steps involved for its solution.because I need its approach to solve it
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3758
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### Re: Integration of log(cos(x))?

Definite integration of log(cos(x)) is very simple to calculate, but what about INDEFINITE Integration,does the solution does not exist in Indefinite integration.
Najam
Newcomer

Posts: 11
Joined: Wed Aug 01, 2012 6:09 am

### Re: Integration of log(cos(x))?

There is no anti-derivative of ln(cos(x)) in elementary terms. One must use the polylogarithmic function.

http://www.wolframalpha.com/input/?i=integral+of+ln%28cos%28x%29%29

I would also ask you to not post the same message more than once in a topic in succession. You have done this several times, even quoting yourself less than 10 minutes after a post only to display the same message. This is unnecessary. Thank you.
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
Global Moderator

Posts: 11318
Joined: Thu Jul 01, 2010 3:14 am
Location: St. Augustine, FL., U.S.A.'s oldest city

### Re: Integration of log(cos(x))?

Najam
Newcomer

Posts: 11
Joined: Wed Aug 01, 2012 6:09 am

### Re: Integration of log(cos(x))?

Yes indeed!
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
Global Moderator

Posts: 11318
Joined: Thu Jul 01, 2010 3:14 am
Location: St. Augustine, FL., U.S.A.'s oldest city

### Re: Integration of log(cos(x))?

Sorry for posting same message in succession.I'll try not to do such thing in future.Thanks for your reply.
Najam
Newcomer

Posts: 11
Joined: Wed Aug 01, 2012 6:09 am

### Re: Integration of log(cos(x))?

A problem closely related to the one posted is:

$\frac{dy}{dx}=x\tan(x)$
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
Global Moderator

Posts: 11318
Joined: Thu Jul 01, 2010 3:14 am
Location: St. Augustine, FL., U.S.A.'s oldest city

### Re: Integration of log(cos(x))?

by separating the variables and using product rule of integration we can get it's solution.if i'm not wrong.
Najam
Newcomer

Posts: 11
Joined: Wed Aug 01, 2012 6:09 am

### Re: Integration of log(cos(x))?

You will wind up with $\int\ln$$\cos(x)$$\,dx$ as one of the terms. I found this related problem when I initially tried integration by parts on the original problem.
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
Global Moderator

Posts: 11318
Joined: Thu Jul 01, 2010 3:14 am
Location: St. Augustine, FL., U.S.A.'s oldest city

### Re: Integration of log(cos(x))?

What geometrically does the given function represents.lim┬(x→2)⁡〖(x^2-4)/(x-2)〗.and during the calculation of limits does that geometry changes or remains the same.because the answer is x+2 which represents a straight line.
Najam
Newcomer

Posts: 11
Joined: Wed Aug 01, 2012 6:09 am

### Re: Integration of log(cos(x))?

Najam wrote:by separating the variables and using product rule of integration we can get it's solution.if i'm not wrong.

Now that's crazytalk!

Saying that an elementary solution to the anti-derivative of ln(cos(x)) exists is equivalent to saying that the solution of x^5+x+a = 0 is expressible in terms of radicals.

Are you trying to disprove (Differential)Galois Theory?!?!
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3758
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### Re: Integration of log(cos(x))?

The function:

$f(x)=\frac{x^2-4}{x-2}$

is equivalent to the function:

$g(x)=x+2$

except at $x=2$ where the former function is undefined...if has a "hole" in the graph there, but otherwise is the same as the latter function.
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
Global Moderator

Posts: 11318
Joined: Thu Jul 01, 2010 3:14 am
Location: St. Augustine, FL., U.S.A.'s oldest city

### First Principle of Derivatives

How can we show derivative of tan(x) using first principle of derivative,graphically?
Najam
Newcomer

Posts: 11
Joined: Wed Aug 01, 2012 6:09 am

Previous