Solving the Integral of cos^2(bx): Is There an Easier Way?

  • Thread starter KingBigness
  • Start date
  • Tags
    Integral
In summary, the integral \int cos^{2}(bx)dx can be solved using the double angle formula to convert the integrand. This results in the equation \cos 2(bx) = 2\cos^2 (bx) -1, which can then be integrated as \frac{1}{2} \int (cos 2(bx)+1)dx. This method is much quicker and only takes 2-3 lines of work.
  • #1
KingBigness
96
0

Homework Statement



[itex]\int cos^{2}(bx)dx[/itex]

Homework Equations





The Attempt at a Solution



This integral popped up in my quantum mechanics class yesterday and I solved it through double substitution to get,
[itex]\frac{x}{2} + sin(bx) * \frac{1}{4b}[/itex].

My question is, someone in my class said there was an easier way to solve this that only took 2-3 lines. Does anyone here have any ideas on what that may be?
 
Physics news on Phys.org
  • #2
Use double angle formula to convert the integrand.
[tex]\cos 2(bx) = 2\cos^2 (bx) -1[/tex]
[tex]\int cos^{2}(bx)dx=\frac{1}{2} \int (cos 2(bx)+1)dx[/tex]
 
  • #3


sharks said:
Use double angle formula to convert the integrand.
[tex]\cos 2(bx) = 2\cos^2 (bx) -1[/tex]
[tex]\int cos^{2}(bx)dx=\frac{1}{2} \int (cos 2(bx)+1)dx[/tex]

Wow that makes it easier...haha always forget my trig identities. Cheers for the quick response.
 

1. What is the integral of cos^2(bx)?

The integral of cos^2(bx) is equal to (1/2)(x + sin(2bx)/2b) + C.

2. Is there an easier way to solve the integral of cos^2(bx)?

Yes, there are various methods for solving the integral of cos^2(bx) such as using trigonometric identities, integration by substitution, or integration by parts.

3. Can the integral of cos^2(bx) be solved without using trigonometric identities?

Yes, the integral can also be solved using integration by substitution or integration by parts.

4. Are there any tips for solving the integral of cos^2(bx) more efficiently?

One tip is to use the trigonometric identity cos^2(x) = (1 + cos(2x))/2 to simplify the integral before solving it. Additionally, using integration tables or online calculators can also help in solving the integral more efficiently.

5. How does the value of b affect the integral of cos^2(bx)?

The value of b affects the integral by changing the frequency and period of the cosine function. This in turn affects the overall shape and amplitude of the integral, but the method for solving it remains the same.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
762
  • Calculus and Beyond Homework Help
Replies
3
Views
344
  • Calculus and Beyond Homework Help
Replies
15
Views
786
  • Calculus and Beyond Homework Help
Replies
4
Views
135
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
22
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
706
  • Calculus and Beyond Homework Help
Replies
11
Views
695
  • Calculus and Beyond Homework Help
Replies
1
Views
492
  • Calculus and Beyond Homework Help
Replies
1
Views
866
Back
Top