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Calculus/Integration problem

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Expert: Alon Mandes - 4/26/2009


Question
How do you find the integral of [sin^2(3x)]*[cos^2(3x)]? (integral sin squared of 3x times cos squared of 3x)

Thanks for the help!

Answer
Let's review some of the facts regarding sin & cos :
1. Cos(2Ø)=2Cos²(Ø)-1  --> Cos²(Ø)=[1+Cos(2Ø)]½
2. Cos(2Ø)=1-2Sin²(Ø)  --> Sin²(Ø)=[1-Cos(2Ø)]½
3. Cos²(Ø)Sin²(Ø)=¼[1+Cos(2Ø)][1-Cos(2Ø)]=¼[1-Cos²(2Ø)]=¼[Sin²(2Ø)]
  =(1/8)[1-Cos(4Ø)] .
So, now let's get back to our problem :

∫Cos²(3x)Sin²(3x) dx = ∫(1/8)[1-Cos(12x)] dx =
(1/8)x-(1/8)∫Cos(12x) dx = x/8 - Sin(12x)/96 .

Alon.  

Calculus

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Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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