Help integrating sin^2(x-pi/6)

In summary, there is a sign error in the end. What is ##\displaystyle \int \cos(x) \, dx##?integrating cos(x) is -sin(x) + c, I took that in consideration as it was 1-cos(x), thus using two negatives = positive, I corrected it to x + 1/2 sin(2x- pi/3).
  • #1
K.QMUL
54
0
Hi there everyone,

sin^2(x-pi/6) dx

I have the following integral to solve but am unsure where I should start, I first thought about integrating by parts as I thought you could split it into [Sin(x-pi/6)][Sin(x-pi/6)]. But couldn't seem to figure that out. I was wondering if you could use a trig identity but again am unsure which one.

Any suggestions?
 
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  • #2
Use the double angle formula. This is a very handy formula to reduce the exponent in a trigfunction appearing in your problem.

Integration by parts works nicely as well, if you are careful with your notation.
 
  • #3
which double angle formula would I use, we still have a Sin^2 to deal with
 
  • #4
Well, use the double angle formula in which sin^2 appears on its own, of course.
 
  • #5
Aha, yes, Thanks for the help
 
  • #6
I have completed the question using the double angle formula, could you tell me if you find any errors, as I am still unsure whether I have done this right or not.

Thanks
 

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  • #7
K.QMUL said:
I have completed the question using the double angle formula, could you tell me if you find any errors, as I am still unsure whether I have done this right or not.

Thanks

There is a sign error in the end. What is ##\displaystyle \int \cos(x) \, dx##?
 
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  • #8
integrating cos(x) is -sin(x) + c, I took that in consideration as it was 1-cos(x), thus using two negatives = positive, I corrected it to x + 1/2 sin(2x- pi/3).

Does everything look good for this integration? Since I checked on Wolfram Mathematica's on-line integration and the answer they got was

http://integrals.wolfram.com/index.jsp?expr=sin^2(x-pi/6)&random=false
 
  • #9
oh, I realize my mistake, integrating cos(x) = sin(x)
 
  • #10
K.QMUL said:
I have completed the question using the double angle formula, could you tell me if you find any errors, as I am still unsure whether I have done this right or not.
You can always check your answer by differentiating it and seeing if you recover the integrand.
 
  • #11
So I've completed the question, and checked if I get the original answer by differentiating it. And it seems good. HOWEVER, I have one concern; when you use the double angle formula, can I take 'A' as (x - pi/6) in sin^2(x-pi/6) or would I need to split it somehow. Please clear up my confusion.
 
  • #12
What does 'A' represent?
 
  • #13
In terms of the question: sin2
 
  • #14
Sorry, in terms of the formula: sin2A = 0.5[1-cos2A]
 
  • #15
In the identity, you can replace 'A' by anything as long as you replace it with the same thing everywhere. Setting A to ##(x-\pi/6)## is perfectly fine.
 

1. What is the basic concept behind "Help integrating sin^2(x-pi/6)"?

The basic concept behind "Help integrating sin^2(x-pi/6)" is to find the integral of the trigonometric function sin^2(x-pi/6), which represents the square of the sine function with a phase shift of pi/6. This involves using various integration techniques and trigonometric identities to simplify the integral and find the final solution.

2. Why is integrating sin^2(x-pi/6) important in science?

Integrating sin^2(x-pi/6) is important in science because the sine function is a fundamental mathematical function that is used to model many natural phenomena, such as sound and light waves. Being able to integrate this function allows us to analyze and understand these phenomena more deeply, and also has practical applications in fields such as engineering and physics.

3. What are the steps for integrating sin^2(x-pi/6)?

The steps for integrating sin^2(x-pi/6) involve using trigonometric identities to rewrite the function in a simpler form, applying integration techniques such as u-substitution or integration by parts, and then using the limits of integration to find the final solution. It may also involve using other integration techniques such as trigonometric substitution or partial fractions.

4. Can you provide an example of how to integrate sin^2(x-pi/6)?

Yes, an example of integrating sin^2(x-pi/6) is as follows:

∫sin^2(x-pi/6)dx = ∫(1-cos(2x-pi/3))/2 dx (using the double angle identity for sine)

= x/2 - (sin(2x-pi/3))/4 + C (using u-substitution with u = 2x-pi/3)

Plugging in the limits of integration (0 and pi) gives the final solution of pi/4.

5. Are there any tips or tricks for integrating sin^2(x-pi/6)?

One tip for integrating sin^2(x-pi/6) is to always check for any possible trigonometric identities or substitutions that can simplify the integral. It can also be helpful to draw a graph of the function to visually understand the integration process. Additionally, practicing various integration techniques and regularly reviewing trigonometric identities can improve the process of integrating sin^2(x-pi/6).

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