# How do you integrate #1/(1+tanx) dx#?

So

Now we perform a partial fraction decomposition on the integrand

So,

and

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Use the substitution

Let

Apply partial fraction decomposition:

Rearrange:

Integrate term by term:

Reverse the substitution:

Simplify:

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To integrate ( \frac{1}{1 + \tan(x)} , dx ), you can use the substitution method.

- Let ( u = 1 + \tan(x) ).
- Find ( du = \sec^2(x) , dx ).
- Rewrite the integral in terms of ( u ): ( \int \frac{1}{u} \cdot \frac{1}{\sec^2(x)} , du ).
- Simplify the expression: ( \int \frac{\cos^2(x)}{u} , du ).
- Integrate ( \frac{1}{u} ) with respect to ( u ): ( \ln|u| + C ).
- Substitute back ( u = 1 + \tan(x) ).

The final result is ( \ln|1 + \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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