Evaluate the integral of $ frac{cos(2x)}{sqrt{1-sin^2(2x)}} $ with respect to $ x $

To evaluate the integral $ \int \frac{\cos(2x)}{\sqrt{1-\sin^2(2x)}} \, dx $, we begin by recognizing that:

$ \sin^2(2x) + \cos^2(2x) = 1 $

Thus, the expression under the square root simplifies to:

$ \sqrt{1-\sin^2(2x)} = \cos(2x) $

Substituting this into the integral gives:

$ \int \frac{\cos(2x)}{\cos(2x)} \, dx $

This simplifies to:

$ \int 1 \, dx $

The integral of 1 with respect to $x$ is:

$ x + C $

To find the integral $ int frac{cos(2x)}{sqrt{1-sin^2(2x)}} , dx $, notice that:

$ cos^2(2x) = 1 – sin^2(2x) $

Thus:

$ sqrt{1-sin^2(2x)} = cos(2x) $

The integral simplifies to:

$ int 1 , dx $

The result is:

$ x + C $

Evaluate the integral $ int frac{cos(2x)}{sqrt{1-sin^2(2x)}} , dx $:

$ sqrt{1-sin^2(2x)} = cos(2x) $

Integral simplifies to:

$ int 1 , dx $

Result:

$ x + C $

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