# How do you find the integral of #1/cos x#?

This is an important trigonometric identity:

Now, use substitution:

Note that these are the fraction's numerator and denominator.

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To find the integral of ( \frac{1}{\cos(x)} ), you can use the substitution method. Let ( u = \tan(\frac{x}{2}) ). Then, ( \cos(x) = \frac{1-u^2}{1+u^2} ), ( dx = \frac{2}{1+u^2} du ).

Substitute these values into the integral:

[ \int \frac{1}{\cos(x)} , dx = \int \frac{1}{\frac{1-u^2}{1+u^2}} \cdot \frac{2}{1+u^2} , du ]

After simplifying, the integral becomes:

[ \int \frac{2}{1-u^2} , du ]

This is a standard integral that can be solved using partial fraction decomposition or recognized as a known integral. The result is:

[ \int \frac{2}{1-u^2} , du = \ln \left| \frac{1+u}{1-u} \right| + C ]

Substituting back ( u = \tan(\frac{x}{2}) ), the final result is:

[ \int \frac{1}{\cos(x)} , dx = \ln \left| \frac{1+\tan(\frac{x}{2})}{1-\tan(\frac{x}{2})} \right| + C ]

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To find the integral of ( \frac{1}{\cos(x)} ), you can use the trigonometric identity ( \sec(x) = \frac{1}{\cos(x)} ). Therefore, the integral becomes the integral of ( \sec(x) ).

The integral of ( \sec(x) ) can be found using integration techniques such as substitution or trigonometric identities. One common method is to use substitution, letting ( u = \tan(x) + \sec(x) ), which simplifies the integral to ( \ln|\sec(x) + \tan(x)| + C ), where ( C ) is the constant of integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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