# How do I evaluate the indefinite integral #int(tan^2(x)+tan^4(x))^2dx# ?

Now, note that

Thus, the substitution

using the power rule for integration (see, e.g, antiderivatives )

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To evaluate the indefinite integral ∫(tan^2(x)+tan^4(x))^2 dx, we can use trigonometric identities to simplify the expression. By using the identity tan^2(x) + 1 = sec^2(x), we can rewrite tan^4(x) as (tan^2(x))^2 = (sec^2(x) - 1)^2. Expanding the square of the sum, we get (tan^2(x) + tan^4(x))^2 = (sec^2(x))^2 - 2sec^2(x) + 1. The integral then becomes ∫((sec^2(x))^2 - 2sec^2(x) + 1) dx. Integrating each term separately, we get ∫(sec^4(x) - 2sec^2(x) + 1) dx. The integral of sec^4(x) can be found using integration by parts or by using the power-reduction formula for integrals of secant functions. After integrating each term, the result will be the evaluated indefinite integral.

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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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