Integral involving trigonometric functions.

In summary, the conversation discusses proving the statement \int_{0}^{\pi}\frac{1-\cos(nx)}{1-\cos(x)} dx=n\pi \ \ , n \in \mathbb{N} using induction and the kernel of Dirichlet. It also mentions the question of whether the statement can be proven by solving the integral.
  • #1
Blandongstein
9
0
I found this question on a website.

Homework Statement


Prove that [itex]\displaystyle \int_{0}^{\pi}\frac{1-\cos(nx)}{1-\cos(x)} dx=n\pi \ \ , n \in \mathbb{N}[/itex]

2. The attempt at a solution

Here's my attempt using induction:

Let [itex]P(n) [/itex] be the statement given by

[tex] \int_{0}^{\pi}\frac{1-\cos(nx)}{1-\cos(x)} dx=n\pi \ \ , n \in \mathbb{N}[/tex]

P(1):

[tex] \int_{0}^{\pi}\frac{1-\cos(x)}{1-\cos(x)} dx=\int_{0}^{\pi}dx=\pi[/tex]
[tex](1)\pi=\pi[/tex]

P(1) holds true.

Let [itex]P(n)[/itex] be true.
Now, we need to show that [itex]P(n+1)[/itex] is true.

[tex]\int_{0}^{\pi} \frac{1-\cos{x(n+1)}}{1-\cos(x)}dx=\int_{0}^{\pi}\frac{1-[\cos(nx)\cos(x)-\sin(nx)\sin(x)]}{1-\cos(x)} dx[/tex]

I don't know how to proceed from here.
Furthermore, I would like to know if I could prove the statement by solving the integral.
 
Physics news on Phys.org
  • #2
Let [itex]P(n)=\int\frac{1-\cos{nx}}{1-\cos{x}}dx[/itex]

[itex]P(n+1)-P(n)=\int\frac{\cos{(n+1)x}-\cos{nx}}{1- \cos{x}}dx=\int\frac{2\sin{(n+\frac{1}{2})x} \sin{\frac{x}{2}}}{2\sin^{2}{\frac{x}{2}}}dx=\int \frac{\sin{(n+\frac{1}{2})x} }{\sin{\frac{x}{2}}}dx[/itex]
(thr limits of integral are 0 and pi)
The last Integral is the kernel of Dirighle and equal to [itex]\pi[/itex].
So [itex]P(n+1)-P(n)=\pi[/itex] .
Finally we obtain [itex]P(n)=n\pi[/itex].
 
Last edited:

1. What is the basic formula for an integral involving trigonometric functions?

The basic formula for an integral involving trigonometric functions is: ∫ sin(x) dx = -cos(x) + C.

2. How do you evaluate an integral involving trigonometric functions?

To evaluate an integral involving trigonometric functions, you can use the trigonometric identities and substitution techniques. First, simplify the integrand using trigonometric identities. Then, substitute a variable for the trigonometric function and solve for the integral using basic integration techniques.

3. What are the common trigonometric identities used in evaluating integrals?

Some common trigonometric identities used in evaluating integrals are: sin²(x) + cos²(x) = 1, sin(x)/cos(x) = tan(x), and cos(x)/sin(x) = cot(x).

4. Can you give an example of an integral involving trigonometric functions?

One example of an integral involving trigonometric functions is: ∫ 2sin(x)cos(x) dx. Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite the integral as: ∫ sin(2x) dx = -1/2 cos(2x) + C.

5. How are integrals involving trigonometric functions used in real-world applications?

Integrals involving trigonometric functions are commonly used in physics and engineering to solve problems related to oscillations, waves, and periodic motion. They are also used in calculating areas and volumes of curved shapes in geometry and in analyzing periodic data in statistics.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
175
  • Calculus and Beyond Homework Help
Replies
11
Views
197
  • Calculus and Beyond Homework Help
Replies
5
Views
596
  • Calculus and Beyond Homework Help
Replies
2
Views
353
  • Calculus and Beyond Homework Help
Replies
3
Views
237
  • Calculus and Beyond Homework Help
Replies
1
Views
56
  • Calculus and Beyond Homework Help
Replies
1
Views
220
  • Calculus and Beyond Homework Help
Replies
6
Views
677
  • Calculus and Beyond Homework Help
Replies
2
Views
836
  • Calculus and Beyond Homework Help
Replies
9
Views
685
Back
Top