# How do you integrate #sqrt(1- tan^2(x))# with respect to #x#?

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Omitting the process of partial fraction decomposition, this becomes:

Both of which are forms of arctangent integrals:

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To integrate ( \sqrt{1 - \tan^2(x)} ) with respect to ( x ), you can use the trigonometric identity ( 1 - \tan^2(x) = \sec^2(x) ). Then, you can rewrite the integral in terms of ( \sec(x) ):

[ \int \sqrt{1 - \tan^2(x)} , dx = \int \sqrt{\sec^2(x)} , dx ]

Now, ( \sqrt{\sec^2(x)} = |\sec(x)| ), because ( \sec(x) ) can be negative in certain intervals. Therefore, we integrate ( |\sec(x)| ) with respect to ( x ):

[ \int |\sec(x)| , dx ]

This integral can be evaluated using techniques such as substitution or trigonometric identities. After integrating, don't forget to include the absolute value sign since ( \sec(x) ) can be negative in some intervals.

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