Integrating sec(x): Understanding the Answer

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In summary, the given conversation is about a problem the person has and how they solved it. The person integrals sec(x) and differentiated it to verify it. They found a solution that was different from what they got. They don't think they made a mistake, but they can't see how their answer is related to the given one. They then summarized the content and explained how they did thedifferentiation and how to evaluate the given integral.
  • #1
ForceBoy
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Hello.

I integrated ##\sec(x) ## and got an answer. I differentiated it to verify it and it came out well. Later when I was looking in a table of integrals I saw the solution ## \ln| \sec(x) +\tan(x) | + c##. This was completely different than my solution. I do not think I made a mistake (unless I'm wrongly convinced of something) and I can't see how my answer is related to the given one.

My work:

##I = \int\sec(x) dx ##

## \sec x = \frac{1}{\cos x}## and ##\cos(x) = \frac{e^{ix}+e^{-ix}}{2}## so

##I = \int \frac{2dx}{e^{ix}+e^{-ix}}##

## I = 2 \int \frac{dx}{e^{ix}+e^{-ix}} ##

Multiplication by ##\frac{e^{ix}}{e^{ix}}## yields

## I = 2 \int \frac{e^{ix}dx}{e^{2ix}+1} ##

Let ##u = e^{ix}##
then ##du = i u dx##
so ##\frac{du}{iu} = dx ##

The integral now:
##I = 2\int \frac{u}{u^{2}+1} \frac{du}{iu}##

##I = -2i\int \frac{1}{u^{2}+1}##

We know ##\int \frac{1}{u^{2}+1} = arctan(x)+c##

So it follows that
##I = -2i \arctan(u) + c##

and finally that

##I = -2i \arctan(e^{ix}) + c##

Differentiation:

## \frac{d}{dx}(-2i \arctan(e^{ix})) ##

## -2i \frac{d}{dx}(\arctan(e^{ix})) ##

##t = ix##
##u = e^{t} ##
##v = \arctan(u)##

so

## -2i ( \frac{dv}{du} * \frac{du}{dt} *\frac{dt}{dx}) ##

##-2i(\frac{1}{u^{2}+1}*u*i) ##

##-2i( \frac{i e^{ix}}{e^{2ix}+1}) ##

## -2i(\frac{i e^{ix}}{e^{ix} (e^{ix} +e^{-ix})})##

##-2i(\frac{i}{e^{ix} +e^{-ix}})##

##\frac{2}{e^{ix} +e^{-ix}}##

##\sec(x)##

This is then clearly different than the mentioned solution. I don't see where I could've made a mistake. If I'm right, my solution would require more simplification. I don't see how this could be done.

Any thoughts would be appreciated. Thanks.
 
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  • #2
##\int \frac{1}{1+u^2}du=arctan(u)+c## holds only for real u. Here ##u=e^{ix}## is complex.
 
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  • #3
I understand. Would my next step then be to evaluate ##\int \frac{1}{z^{2} +1} dz ## where z is a complex number?
 
  • #4
Also a question.
mathman said:
##\int \frac{1}{1+u^2}du=arctan(u)+c## holds only for real u. Here ##u=e^{ix}## is complex.

Why is this? In the differentiation of arctan(x), where is it that it is assumed that x is real?
 
  • #5
Technically you are correct. However arctan(u) when u is complex is not very useful. It is more useful with the form given (ln|...|)..
 
  • #6
I now saw how to evaluate ## \int \frac{du}{u^2+1} ## for complex u. It's ## \frac{i}{2}\ln|\frac{u+i}{u-i}| ## if I'm not wrong. If I use this instead of ##arctan(u)## I get:

## I = \ln|\frac{e^{ix}+i}{e^{ix}-i}| ##

When looking at this, something tells me it can quickly be expressed as ## \ln|\sec(x)+\tan(x)| ##. Is this right?
 
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  • #7
I just found out how to make it ## \ln|\sec(x)+\tan(x)| ##. I now understand this better and know how to integrate sec(x). Thank you for your help and time. :smile:
 

Related to Integrating sec(x): Understanding the Answer

1. What is the meaning of "integrating sec(x)"?

Integrating sec(x) refers to the mathematical process of finding the antiderivative of the secant function, which is denoted as ∫sec(x) dx. This involves finding a function whose derivative is equal to sec(x).

2. How is the integral of sec(x) evaluated?

The integral of sec(x) can be evaluated using the substitution method or integration by parts. In the substitution method, we substitute u = tan(x/2) and rewrite sec(x) in terms of u. In integration by parts, we use the formula ∫u dv = uv - ∫v du with u = sec(x) and dv = dx.

3. What is the result of integrating sec(x)?

The result of integrating sec(x) is ln|sec(x) + tan(x)| + C, where C is a constant of integration. This is derived from the integration formula of sec(x) and using the trigonometric identity sec(x) = 1/cos(x).

4. Can the integral of sec(x) be simplified further?

The integral of sec(x) cannot be simplified further in terms of elementary functions. However, it can be expressed in terms of special functions such as the logarithmic integral, which is defined as Li(x) = ∫1/x ln(t) dt.

5. How is the integral of sec(x) used in real-life applications?

The integral of sec(x) has various applications in physics, engineering, and economics. For example, it is used in calculating the work done by a force acting on an object, determining the center of mass of a system of particles, and finding the area under a curve in economic analysis.

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