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To integrate (\sin(2\omega t)), we use the integral formula for sine. The integral is given by:
[ \int \sin(2\omega t) , dt = -\frac{1}{2\omega} \cos(2\omega t) + C ]
where (C) is the constant of integration. The factor of (-\frac{1}{2\omega}) comes from the chain rule, accounting for the derivative of (2\omega t).
What else can I help you with?
How many half steps is a major third made of?
Four half steps, or two whole steps.
How many half steps are in a tritone?
6 half steps
How many half steps are in an augmented triad?
An augmented triad consists of two major thirds stacked on top of each other. In terms of half steps, the interval from the root to the major third is four half steps, and the interval from the major third to the augmented fifth is another four half steps. Therefore, an augmented triad contains a total of eight half steps from the root to the highest note.
How do you integrate cos squared theta times sine theta?
To integrate ( \cos^2 \theta \sin \theta ), you can use a substitution method. Let ( u = \cos \theta ), then ( du = -\sin \theta , d\theta ). The integral becomes ( -\int u^2 , du ), which evaluates to ( -\frac{u^3}{3} + C ). Substituting back, the final result is ( -\frac{\cos^3 \theta}{3} + C ).
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