Find the integral of 2sin(x)cos(x) dx.

To find the integral of (2\sin(x)\cos(x) , dx), we can use a well-known trigonometric identity. Specifically, the double angle identity states that:

[

\sin(2x) = 2\sin(x)\cos(x).

]

Using this identity, we can rewrite the integral as:

[

\int 2\sin(x)\cos(x) , dx = \int \sin(2x) , dx.

]

The integral of (\sin(2x)) can be solved using a substitution. If we let (u = 2x), then (du = 2dx) or (dx = \frac{du}{2}). Substituting into the integral gives:

[

\int \sin(2x) , dx = \frac{1}{2} \int \sin(u) , du.

]

The integral of (\sin(u)) is (-\cos(u) + C), so we have:

[

\frac{1}{2}(-\cos(u)) + C = -\frac{1}{2}\cos(2x) + C.

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