Trig Integration Question

As Dick suggested, try to derive them from the integration by parts formula.In summary, the conversation discusses the integral \int^{\pi}_{0}(cos(x))^{6}dx and different methods for solving it. One method is to use the half-angle formula for cos(x)^2 and apply it multiple times, but it can lead to a messy solution. Another method is to use a reduction formula, which involves expressing the integral in terms of an integral with a lower power of cos(x). The conversation also mentions using hyperbolic cosine to simplify the integral, but it is not recommended. Finally, the conversation mentions two cases for solving integrals involving powers of cosine and sine, and suggests deriving the reduction formulas from the integration
  • #1
28
0

Homework Statement



[tex]\int^{\pi}_{0}(cos(x))^{6}dx[/tex]

Homework Equations



* Half-Angle => (cos(x))^{2} = (1/2)(1 + cos(2x))

The Attempt at a Solution



We just started this chapter today and during lecture the only example of this form (even powers/cosine) was (cos(x))^{2}, which only requires integrating the Half-Angle Formula. The way I approached this problem looks like it's taking me towards pretty big mess:

[tex]\int^{\pi}_{0}((cos(x))^{2})^{3}[/tex] = [tex]\int^{\pi}_{0}[(1/2)(1+cos(2x))]^{3}[/tex]

Any suggestion?
 
Physics news on Phys.org
  • #2
How about using hyperbolic cosh to get the things into full to e's?
 
  • #3
The way to avoid the mess is to use a formula to express the integral of cos^n(x) in terms of an integral of cos^(n-2)(x). See for example http://www.sosmath.com/calculus/integration/powerproduct/powerproduct.html at the bottom of the page. You should also note that you have a DEFINITE integral from 0 to pi. You don't need to evaluate the full indefinite integral. The term involving sin(x) in the formula vanishes in your case. If you don't already know that formula, you should probably try to prove it. It's not too hard. Just integration by parts.
 
  • #4
crm08 said:

Homework Statement



[tex]\int^{\pi}_{0}(cos(x))^{6}dx[/tex]

Homework Equations



* Half-Angle => (cos(x))^{2} = (1/2)(1 + cos(2x))

The Attempt at a Solution



We just started this chapter today and during lecture the only example of this form (even powers/cosine) was (cos(x))^{2}, which only requires integrating the Half-Angle Formula. The way I approached this problem looks like it's taking me towards pretty big mess:

[tex]\int^{\pi}_{0}((cos(x))^{2})^{3}[/tex] = [tex]\int^{\pi}_{0}[(1/2)(1+cos(2x))]^{3}[/tex]

Any suggestion?
Tedious but not that big a mess:
[tex]=\frac{1}{8} \int_0^\pi 1+ 3cos(2x)+ 3cos^2(2x)+ cos^3(2x) dx[/tex]
you can immediately integrate [itex]\int 3cos(2x) dx[/itex] and writing [itex]cos^3(2x)[/itex]as [itex]cos^2(2x)cos(x)= (1- sin^2(x))cos(x)[/itex] let's you integrate \(\displaystyle \int cos^3(2x)dx[/itex].

The only "hard" part is
[tex](3/8)\int_0^\pi cos^2(2x)dx[/tex]
and you can use [tex]cos^2(2x)= (1/2)(1+ cos(4x))[/tex] for that.\)
 
  • #5
My prof covered this whole chapter today and only introduced two "cases" that can be identified for solving these problems, each case having two methods which are chosen by looking at the even or odd powers.

case 1: integral[((cosx)^m)*(sinx)^n))dx]

- or -

case 2: integral[((tanx)^n)*((secx)^m)dx]

I figured these were the only ways to solve these problems but I'm open to any new methods. I'm not clear on what you are telling me to do, are you saying to somehow get it in the form:

(e^x+e^x)/2

**sorry this reply was to "rootx", it took me awhile to post I didn't see the other replies after
 
  • #6
Dick, thanks for the link, I forgot about that reduction formula.
 
  • #7
crm08 said:
My prof covered this whole chapter today and only introduced two "cases" that can be identified for solving these problems, each case having two methods which are chosen by looking at the even or odd powers.

case 1: integral[((cosx)^m)*(sinx)^n))dx]

- or -

case 2: integral[((tanx)^n)*((secx)^m)dx]

I figured these were the only ways to solve these problems but I'm open to any new methods. I'm not clear on what you are telling me to do, are you saying to somehow get it in the form:

(e^x+e^x)/2

**sorry this reply was to "rootx", it took me awhile to post I didn't see the other replies after

I was thinking of:
(cos (x))^6 = [(exp(i.x) + exp(-i.x))/2]^6
It's easier to expand the right side and not hard to integrate.
I remember using this several times somewhere but I have forgotten where:frown:

But, those reductions formulas are best here.
 

1. What is trigonometric integration?

Trigonometric integration is the process of finding the integral of a trigonometric function. This involves using techniques such as substitution, integration by parts, and trigonometric identities.

2. How do you integrate trigonometric functions?

To integrate a trigonometric function, you need to first identify which integration technique would be most appropriate. Then, use the appropriate technique to manipulate the function into a form that can be easily integrated. Finally, solve the integral and check your solution using differentiation.

3. What are some common trigonometric identities used in integration?

Some common trigonometric identities used in integration include the Pythagorean identities, double angle identities, and half-angle identities. These identities can help simplify trigonometric functions and make them easier to integrate.

4. Can you explain the concept of u-substitution in trigonometric integration?

U-substitution is a technique used in integration where you substitute a part of the function with a new variable, u, to make the integral easier to solve. In trigonometric integration, this often involves substituting a trigonometric function with its derivative or using trigonometric identities to rewrite the function in terms of u.

5. How can I practice and improve my skills in trigonometric integration?

The best way to practice and improve your skills in trigonometric integration is to solve a variety of problems. You can find practice problems in textbooks, online resources, or create your own by coming up with different types of trigonometric functions to integrate. Additionally, it can be helpful to review the properties and techniques of integration, as well as the common trigonometric identities used in integration.

Suggested for: Trig Integration Question

Replies
9
Views
1K
Replies
30
Views
288
Replies
8
Views
511
Replies
10
Views
226
Replies
5
Views
698
Replies
9
Views
733
Replies
7
Views
630
Replies
1
Views
670
Back
Top