# How do I find the antiderivative of #f(x)=tan(2x) + tan(4x)#?

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Integration is also known as the anti derivative.

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To find the antiderivative of ( f(x) = \tan(2x) + \tan(4x) ), use the trigonometric identity (\tan(a) = \frac{{\sin(a)}}{{\cos(a)}}). Then integrate each term separately.

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

For ( \tan(4x) ): [ \int \tan(4x) , dx = -\frac{1}{4} \ln|\cos(4x)| + C ]

So, the antiderivative of ( f(x) ) is: [ \int \left( \tan(2x) + \tan(4x) \right) , dx = -\frac{1}{2} \ln|\cos(2x)| - \frac{1}{4} \ln|\cos(4x)| + 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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