Trigonmetric Integration Question

In summary, Nick is a new member of the forum who is currently taking a gap year before attending university. He has been using his free time to learn Calculus II topics and has encountered an integral that he is unsure how to properly integrate. After some calculations and referencing the correct answer, he realizes that his mistake was not recognizing that tanx is the primitive of sec²x. He apologizes for the confusion caused by using a different tex protocol and acknowledges that trigonometric identities can be challenging.
  • #1
calcnd
20
0
Hi, I'm new to this forum. I figure it would be appropriate to briefly introduce myself before asking my question.

My name is Nick. I've graduated high school and I am presently taking the year off before attending university next fall. Having a lot of free time has allowed me to begin learning Calculus II topics that I'll inevitably encounter in the future.

Anyways, on to my question(s):

[tex]\int {{-1}\over{cos^2(x)}} dx[/tex]
[tex]= -\int {{sin^2(x) + cos^2(x)}\over{cos^2(x)}} dx [/tex]
[tex]= -\int {{sin^2(x)}\over{cos^2(x)}} dx -\int {{cos^2(x)}\over{cos^2(x)}}dx [/tex]
[tex]= -\int tan^2(x) dx - \int dx [/tex]
[tex]= ?[/tex]

Alas, I do not know how to properly integrate [tex]tan^2(x)[/tex]

I am relatively confident, however, that my above reasoning is corect. This is because the answer to the following integral is supposed to be:

[tex]\int tan^2(x)dx = tan(x) - x + C[/tex] Meaning:

[tex]\int {{-1}\over{cos^2(x)}} dx[/tex]
[tex]= -\int tan^2(x) dx - \int dx [/tex]
[tex]= -tan(x) + x - C - x + D [/tex]
[tex]= -tan(x) + K [/tex]


Which should be the correct answer. Thus, my problem is in integrating the tangent x squared function.Thank-you in advance,

Nick

Edit: It appears this forum uses a different tex protocol than what I am used to on another forum I frequent. I'm attempting to figure this out, but I do appologize for the confusing mess above.

Edit 2: I believe I found the culprit. The textit code does not apply on this forum. :)
 
Last edited:
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  • #2
[tex] \int \tan^{2}x dx= \int \sec^{2}x-1\ dx = \tan x - x + C [/tex]
 
Last edited:
  • #3
Ugh... now don't I feel rather stupid.

Trigonmetric identities will be the death of me.
 
  • #4
courtrigrad said:
[tex] \int \tan^{2}x dx= \int \sec^{2}x-1\ dx = \tan x - x + C [/tex]

Yeah but [tex]\int \sec^2x dx[/tex] is precisely what he's trying to integrate. The question is, how can it be seen that tanx is a primitive of sec²x?
 
  • #5
I guess you're just assumed to have differentiated tanx before and you remembered that is gave sec²x.
 
  • #6
Yep,

I knew [tex]{d\over{dx}} tanx = sec^2x [/tex]. :)
 

Related to Trigonmetric Integration Question

1. What is Trigonometric Integration?

Trigonometric integration is a method used to find the integral of a function involving trigonometric functions such as sine, cosine, tangent, etc. It involves using trigonometric identities and substitution techniques to solve the integral.

2. Why is Trigonometric Integration important?

Trigonometric integration is important because it allows us to solve integrals involving trigonometric functions, which are commonly used in many mathematical and scientific applications. It also helps in simplifying complex integrals and makes them easier to solve.

3. What are some common techniques used in Trigonometric Integration?

Some common techniques used in Trigonometric Integration include trigonometric identities, substitution, integration by parts, and partial fractions. These techniques help in simplifying the integral and making it easier to solve.

4. What are some tips for solving Trigonometric Integration problems?

Some tips for solving Trigonometric Integration problems include identifying the appropriate trigonometric identity to use, choosing the right substitution, and simplifying the integrand as much as possible before integrating. It is also important to be familiar with the properties of trigonometric functions and their derivatives.

5. Can Trigonometric Integration be used in real-life applications?

Yes, Trigonometric Integration can be used in various real-life applications such as physics, engineering, and economics. For example, it can be used to determine the displacement, velocity, and acceleration of an object in motion, or to calculate the area under a curve in a business or economic model.

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