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

We have:

Thus:

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To integrate (\int \frac{x}{\cos^2(x)} , dx), you can use the substitution method. Let (u = \tan(x)), then (du = \sec^2(x) , dx). Rewrite the integral in terms of (u):

(\int \frac{x}{\cos^2(x)} , dx = \int \frac{x}{1 + \tan^2(x)} , dx = \int \frac{x}{1 + u^2} , du)

Now, we can integrate (\int \frac{x}{1 + u^2} , du) using integration by parts, where (dv = \frac{1}{1 + u^2} , du) and (u = x):

(\int \frac{x}{1 + u^2} , du = x \arctan(u) - \int \arctan(u) , dx)

Integrating (\int \arctan(u) , du) gives (u \arctan(u) - \frac{1}{2} \ln(1 + u^2) + C), where (C) is the constant of integration.

Therefore, the final result is:

(\int \frac{x}{\cos^2(x)} , dx = x \arctan(\tan(x)) - \frac{1}{2} \ln(1 + \tan^2(x)) + C)

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