Find Integral of sin(2θ) dθ

To find the integral of sin(2θ) dθ, we can use the standard formula for integrating sine functions. The integral we want to evaluate is:

∫ sin(2θ) dθ

To solve this, we can use a substitution method. Let’s let:

u = 2θ

Then, the differential du will be:

du = 2 dθ → dθ = (1/2) du

Now we can rewrite the integral in terms of u:

∫ sin(2θ) dθ = ∫ sin(u) (1/2) du = (1/2) ∫ sin(u) du

The integral of sin(u) is:

∫ sin(u) du = -cos(u) + C

So we have:

(1/2) ∫ sin(u) du = (1/2)(-cos(u) + C) = -1/2 cos(2θ) + C

Finally, substituting back for u = 2θ gives us the final result:

∫ sin(2θ) dθ = -1/2 cos(2θ) + C

Where C is the constant of integration. Therefore, the integral of sin(2θ) dθ is:

-1/2 cos(2θ) + C