Compute the integral ∫ (sin(x) / cos^2(x)) dx.

To evaluate the integral of (\int \frac{\sin(x)}{\cos^2(x)} , dx), we can recognize that it can be simplified using substitution. The expression (\frac{\sin(x)}{\cos^2(x)}) can be rewritten as (\sin(x)\cdot\sec^2(x)), which suggests that a trigonometric identity or a substitution would be effective.

Let’s use the substitution:

• Let (u = \cos(x)). Then, the derivative (du = -\sin(x) , dx), or equivalently, (dx = \frac{-du}{\sin(x)}).

Now, substituting (\sin(x) = -du) and (\cos^2(x) = u^2), we can transform the integral:

[

\int \frac{\sin(x)}{\cos^2(x)} , dx = \int \frac{-du}{u^2} = -\int u^{-2} , du

]

Integrating (u^{-2}) gives us:

[

-(-u^{-1}) = \frac{1}{u} + C = \frac{