# How do you integrate #int x/ cos^2(x) dx#?

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After using a basic trig identity, this integral becomes a simple case of integration by parts.

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To integrate ( \frac{x}{\cos^2(x)} ) with respect to ( x ), you can use integration by parts method. Let ( u = x ) and ( dv = \frac{1}{\cos^2(x)}dx ).

[ du = dx ]

[ v = \int \frac{1}{\cos^2(x)} dx = \tan(x) ]

Now apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

[ \int \frac{x}{\cos^2(x)} dx = x \tan(x) - \int \tan(x) dx ]

Now integrate ( \tan(x) ) using the formula:

[ \int \tan(x) dx = -\ln|\cos(x)| + C ]

Thus, the final result is:

[ \int \frac{x}{\cos^2(x)} dx = x \tan(x) + \ln|\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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