Integral of Secant: Solving for \int \frac{1}{cosx} dx | Step-by-step Solution

In summary: The integral is the inverse of the derivative, so it's a little more complicated.In summary, the conversation revolves around finding the integral of 1/cosx using a substitution method. The attempt at a solution involves factoring out a negative sign in front of the substitution variable, which leads to an incorrect solution. It is then corrected by noting the negative sign and using the proper integration formula. The conversation also briefly touches on the difference between the derivative and integral of secant.
  • #1
Sisyphus
62
0

Homework Statement



Basically, I have to find

[tex]
\int \frac{1}{cosx} dx
[/tex]

by multiplying the integrand by [tex] \frac{cosx}{cosx}[/tex]

I go through and arrive at a solution, but when I differentiate it,
I get -tan(x)

something's clearly wrong, but I can't see what it is that I'm doing wrong here...

Homework Equations



[tex]
let u = sin(x)[/tex][tex] du = cos(x)dx
[/tex]

The Attempt at a Solution



[tex]
\int \frac{1}{cosx} dx = \int \frac{cosx}{cos^2x} dx\\
= \int \frac{cosx}{1-sin^2x} dx\\
=\int \frac{du}{1-u^2} \\
=\int \frac{du}{(1-u)*(1+u)} \\
=\frac{1}{2} * \int \frac {1}{1+u} + \frac {1}{1-u} du\\
= \frac{1}{2} * (ln(1-u^2}})
=\frac{1}{2} * (ln(cos^2))[/tex]
 
Last edited:
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  • #2
Your second to last step (where you actually perform the integration) is wrong. In the second term, remember that u has a minus sign.
 
  • #3
hmm..

I've never run into anything like this before

so why does u having a minus sign in front of it pose a problem with what I did in my original solution?

(thanks for the help)
 
  • #4
Well, the value of your integral will be ln(1+u)-ln(1-u)
 
  • #5
if I kept the minus sign where it was, which is what I did in my original solution, my integral would've been
[tex] ln(1+u)+ln(1-u)=ln(1-u^2)[/tex]

StatusX told me to watch out for the negative sign in front of the u, so I factored it out, made sure that u was positive, and then integrated it, which gave me the correct solution.

the thing is I'm not sure why I couldn't proceed as usual with the minus sign in front of the u
 
  • #6
StatusX said to note the minus sign in front of the u. You can proceed as normal, but noting that [tex]\int\frac{1}{1-u}du=-ln(1-u)[/tex]. In general [tex]\int\frac{1}{f(u)}du=\frac{ln[f(u)]}{df/du}[/tex]. In this case, f(u)=1-u, and so df/du=-1
 
  • #7
ah, ok

thank you!
 
  • #8
o_O i didn't know we would utilizie such a method to do this integral.

i've always thought the integral of secant was just sec[x]tan[x]!
 
  • #9
silver-rose said:
o_O i didn't know we would utilizie such a method to do this integral.

i've always thought the integral of secant was just sec[x]tan[x]!

That's the derivative of sec(x)
 

1. What is the definition of the integral of secant?

The integral of secant is a mathematical concept that represents the area under the curve of the secant function. It is denoted by ∫ sec(x) dx and is a fundamental integral in calculus.

2. How is the integral of secant calculated?

The integral of secant can be calculated using integration techniques such as substitution, integration by parts, or trigonometric identities. It is a complex integral and may require multiple steps to solve.

3. What is the significance of the integral of secant in mathematics?

The integral of secant is important in mathematics as it helps in solving various real-world problems involving curves and areas. It is also used in various fields such as physics, engineering, and economics.

4. Can the integral of secant be evaluated using a calculator?

Yes, some calculators have the capability to evaluate integrals, including the integral of secant. However, it is important to understand the concept and steps to solve the integral manually before relying on a calculator.

5. Are there any applications of the integral of secant in everyday life?

Yes, the integral of secant has many practical applications, such as calculating the amount of material needed to construct a curved structure, determining the volume of an irregularly shaped object, and finding the distance traveled by a moving object with varying speed.

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