Evaluate the integral of sec^2(x) dx.

The integral of sec^2(x) dx is tan(x) + C, where C is the constant of integration.

To evaluate the integral of sec^2(x) dx, we can use the formula for the integral of a power function. Specifically, we can use the formula:

∫x^n dx = (x^(n+1))/(n+1) + C

where C is the constant of integration. We can apply this formula by letting n = -2, since sec^2(x) is equal to (1/cos^2(x)), which can be written as cos^(-2)(x). Then we have:

∫sec^2(x) dx = ∫cos^(-2)(x) dx
= (cos^(-2)(x) + 1) dx
= cos^(-2)(x) dx + ∫dx
= cos^(-2)(x) dx + x + C

Now we need to evaluate the integral of cos^(-2)(x) dx. We can do this by using the substitution u = sin(x), which gives us:

cos^(-2)(x) dx = (1 + tan^2(x)) dx
= (1 + u^2)/(1 - u^2) du
= (1/(1 - u^2)) du + (u^2/(1 - u^2)) du
= arctan(u) + ln|1 - u^2| + C
= arctan(sin(x)) + ln|cos(x)| + C

Substituting this back into our original integral, we get:

∫sec^2(x) dx = cos^(-2)(x) dx + x + C
= arctan(sin(x)) + ln|cos(x)| + x + C
= tan(x) + C

Therefore, the integral of sec^2(x) dx is tan(x) + C.