How to Integrate sin2x?

To integrate the function sin(2x), we can use a basic rule of integration along with a substitution method. The integral we need to solve is:

∫ sin(2x) dx

First, let’s make a substitution to simplify the integration. We’ll let:

u = 2x

Then, taking the derivative of u with respect to x gives us:

du/dx = 2

This means:

dx = du/2

Now we can substitute u and dx back into the integral, yielding:

∫ sin(u) (du/2)

This simplifies to:

1/2 ∫ sin(u) du

The integral of sin(u) is a standard result:

∫ sin(u) du = -cos(u) + C

So we have:

1/2 (-cos(u)) + C = -1/2 cos(u) + C

Finally, we substitute back u = 2x to get the final result:

∫ sin(2x) dx = -1/2 cos(2x) + C

And that’s the integral of sin(2x). The constant C represents an arbitrary constant of integration.