Final answer:
To integrate sin^2(1-2x/3), use the trigonometric identity that converts sin^2(theta) into a function of cosine, integrate using substitution, and add the constant of integration.
Step-by-step explanation:
Integrating ´sin^2(1-2x/3)
To integrate the function sin^2(1-2x/3), we use a trigonometric identity that relates the square of sine to a function involving cosine. The identity used is sin^2(θ) = (1 - cos(2θ))/2, which allows us to express the squared sine function in terms of cosine, making the integration simpler.
Applying this identity, you get:
1. sin^2(1-2x/3) = (1 - cos(2(1-2x/3)))/2
2. sin^2(1-2x/3) = (1 - cos(2 - 4x/3))/2
Now, you can integrate this function more easily. Remember to apply the Chain Rule when taking the antiderivative of the cosine term.
For the antiderivative, let's say ∫ cos(2-4x/3) dx, use a substitution u = 2 - 4x/3, du = -4/3 dx.
After completing the integration, don't forget to undo the substitution to express the antiderivative in terms of x and add the constant of integration, C. This process is typical in trigonometric integrals encountered in calculus. As for the reference information given, it appears unrelated to the question about integrating sin^2(1-2x/3) and seems to pertain to physics and statics problems involving torques, forces, and trigonometric functions, which may involve integration but are contextually different from the problem at hand.