# How do you integrate #tanx / (secx + cosx)#?

We should first try to simplify the integrand.

Thus:

This is the arctangent integral!

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To integrate ( \frac{\tan(x)}{\sec(x) + \cos(x)} ), you can use the substitution method. Let ( u = \sec(x) + \cos(x) ). Then, ( du = (\sec(x)\tan(x) - \sin(x))dx ).

Now, substitute ( u ) and ( du ) into the integral:

[ \int \frac{\tan(x)}{\sec(x) + \cos(x)} , dx = \int \frac{1}{u} , du ]

This integral is straightforward to solve:

[ \int \frac{1}{u} , du = \ln|u| + C ]

Substitute ( u = \sec(x) + \cos(x) ) back into the result:

[ \ln|\sec(x) + \cos(x)| + C ]

Therefore, the integral of ( \frac{\tan(x)}{\sec(x) + \cos(x)} ) is ( \ln|\sec(x) + \cos(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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