Very Difficult Trig Sub?

  • Thread starter n4rush0
  • Start date
  • Tags
    Trig
In summary: Thanks!In summary, the student is trying to solve the problem using trig substitution and partial fractions.
  • #1
14
0

Homework Statement


Integral of 1/(2+sin(x)) dx


Homework Equations





The Attempt at a Solution


I've been told that you can use trig subs, but I never had to learn that in high school and it hasn't appeared in any of my calculus coursework.

As a side note. I've been wondering if it is possible to solve asin(x) + bcos(x) = c
 
Physics news on Phys.org
  • #2
To solve your side note use this website. it helped me out! good luck

http://www.education2000.com/demo/demo/btnchtml/sinplcos.htm [Broken]
 
Last edited by a moderator:
  • #3
Wow, that's so cool. Thanks for the link. I'll try to remember how to derive it.
 
  • #4
n4rush0 said:

Homework Statement


Integral of 1/(2+sin(x)) dx

The standard substitution [itex] \tan\frac{x}{2} = t [/itex] applies for your integral. It will convert it into a integral of an algebraic function for which the method of partial fraction decomposition will get it solved.
 
  • #5
Thanks, I'll try that. Is that something you just memorized or is there a certain rule that let's you know what to substitute?
 
  • #6
There are rules. That substitution will apply to an antiderivative of a function a+b\sin x/c+d\cos x and the other 3 ways of interchanging cos with sin and more generally to any algebraic function of sin and cos.
 
  • #7
Where can I learn all these rules? I usually only see substitutions with x = asint, atant, or asect
 
  • #8
You're normally taught these rules of substitution in high-school. I wasn't, so I picked them up for myself from books, especially for engineers, because the proofs are missing :)
 
  • #9
Okay so, given:
integral dx/(2+sinx)

tan(x/2) = t
(1/2)sec^2 (x/2) dx = dt
dx = 2cos^2 (x/2) dt

integral
2cos^2 (x/2) dt / (2+sinx)

Am I supposed to use x = arctan(2t)? If so, is it possible to simplify by drawing a triangle?
 
  • #10
Of course you have to use that. It's the whole purpose of substitution, you need to change every function of x including the dx with the approproate function of t and dt.
 
  • #11
I know how to change sinx to sin 2t/sqrt(1+4t^2)
but I'm not sure how to simplify cos^2 (x/2) since it has the 1/2 in front of the x and I can't use the same trick that I used for sinx.
 
  • #12
But you need sin (2 arctan t) from the initial integral.

[tex] \sin (2\arctan t) = 2 (\sin\arctan t) (\cos\arctan t) [/tex]

[tex] \sin\arctan t = \frac{t}{\sqrt{1+t^2}} \, , \, \cos\arctan t = \frac{1}{\sqrt{1+t^2}} [/tex]

What about the integration element ?
 
  • #13
n4rush0 said:
Okay so, given:
integral dx/(2+sinx)

tan(x/2) = t
(1/2)sec^2 (x/2) dx = dt
dx = 2cos^2 (x/2) dt

integral
2cos^2 (x/2) dt / (2+sinx)

Am I supposed to use x = arctan(2t)? If so, is it possible to simplify by drawing a triangle?

It's not x=arctan(2t), but rather x=2arctan(t). that might help.
 
  • #14
Thank you. I modified the integral to
dt/t^2+t+1)
Are you sure it's partial fractions?
 
  • #15
There, I would actually use completing the square in the denominator, then do another trig sub.
 
  • #16
Thank you. I finally get it now. I'll still have problems with the initial trig substitutions though since I'm not sure how to get tan(x/2) = t.
 
  • #17
Letting [itex]t = \tan(x/2)[/itex] is part of something called a Weierstrass substitution. This is usually a pretty messy substitution, but it's good to have in your toolbox of integration tricks, especially for those pesky integrals where nothing else seems to work.
 

What is Trigonometric Substitution?

Trigonometric substitution is a technique used in calculus to simplify integrals involving expressions that contain trigonometric functions.

Why is Trigonometric Substitution Considered Difficult?

Trigonometric substitution can be difficult because it requires a good understanding of trigonometric identities and the ability to manipulate them in various ways.

When Should Trigonometric Substitution be Used?

Trigonometric substitution is typically used when an integral contains a radical expression that cannot be easily simplified or when an integral involves a quadratic expression that cannot be factored.

What are the Most Commonly Used Trigonometric Substitutions?

The most commonly used trigonometric substitutions are the Pythagorean identities, the double angle identities, and the half angle identities.

How Can I Improve My Skills in Solving Difficult Trigonometric Substitutions?

To improve your skills in solving difficult trigonometric substitutions, practice solving a variety of integrals and familiarize yourself with common trigonometric identities and their applications.

Suggested for: Very Difficult Trig Sub?

Replies
3
Views
921
Replies
7
Views
645
Replies
1
Views
676
Replies
3
Views
883
Replies
9
Views
1K
Replies
2
Views
686
Back
Top