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

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

• A. -sec(x) + C
• B. sec(x) + C
• C. -tan(x) + C
• D. sin(x) + C

Answer: (A)

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{

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{