Key Concepts and Formulas

• Trigonometric Identities: $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$, $\sec 2\theta = \frac{1 + \tan^2 \theta}{1 - \tan^2 \theta}$, $\sec^2 \theta = 1 + \tan^2 \theta$
• Substitution Method for Integration: If $\int f(g(x))g'(x) dx$, substitute $t = g(x)$, then $dt = g'(x) dx$, and the integral becomes $\int f(t) dt$.
• Partial Fraction Decomposition (although not strictly needed, understanding the underlying principle is important)

Step-by-Step Solution

Step 1: Rewrite the integral using trigonometric identities. We start with the given integral: $I = \int \frac{d\theta}{\cos^2 \theta (\tan 2\theta + \sec 2\theta)}$ We substitute the double angle formulas for $\tan 2\theta$ and $\sec 2\theta$: $I = \int \frac{d\theta}{\cos^2 \theta \left(\frac{2\tan \theta}{1 - \tan^2 \theta} + \frac{1 + \tan^2 \theta}{1 - \tan^2 \theta}\right)}$

Step 2: Simplify the expression inside the integral. $I = \int \frac{d\theta}{\cos^2 \theta \left(\frac{2\tan \theta + 1 + \tan^2 \theta}{1 - \tan^2 \theta}\right)} = \int \frac{(1 - \tan^2 \theta) d\theta}{\cos^2 \theta (1 + \tan \theta)^2}$ Since $\sec^2 \theta = \frac{1}{\cos^2 \theta}$, we have: $I = \int \frac{(1 - \tan^2 \theta) \sec^2 \theta d\theta}{(1 + \tan \theta)^2}$

Step 3: Use substitution. Let $t = \tan \theta$. Then, $dt = \sec^2 \theta d\theta$. Substituting these into the integral gives: $I = \int \frac{(1 - t^2)}{(1 + t)^2} dt$

Step 4: Factor and simplify the integrand. $I = \int \frac{(1 - t)(1 + t)}{(1 + t)^2} dt = \int \frac{1 - t}{1 + t} dt$

Step 5: Rewrite the integrand to make it easier to integrate. We can rewrite $\frac{1 - t}{1 + t}$ as follows: $\frac{1 - t}{1 + t} = \frac{-(t + 1) + 2}{1 + t} = -1 + \frac{2}{1 + t}$ Therefore, $I = \int \left(-1 + \frac{2}{1 + t}\right) dt$

Step 6: Integrate. $I = \int -1 dt + \int \frac{2}{1 + t} dt = -t + 2\ln|1 + t| + C$

Step 7: Substitute back for $\theta$. Replace $t$ with $\tan \theta$: $I = -\tan \theta + 2\ln|1 + \tan \theta| + C$

Step 8: Compare with the given form. We are given that the integral is of the form $\lambda \tan \theta + 2\log_e |f(\theta)| + C$. Comparing this with our result, we have: $\lambda = -1$ $f(\theta) = 1 + \tan \theta$

Therefore, the ordered pair $(\lambda, f(\theta))$ is $(-1, 1 + \tan \theta)$.

Common Mistakes & Tips

• Trigonometric Identities: Make sure to use the correct trigonometric identities, especially the double-angle formulas.
• Algebraic Manipulation: Be careful with the algebraic manipulation of the integrand to ensure that the integration can be performed easily.
• Substitution: Remember to substitute back the original variable after integration.

Summary

We started by simplifying the given integral using trigonometric identities and then used a substitution to convert the integral into a simpler form. After integrating the simplified expression, we substituted back to obtain the final result in terms of $\theta$. Finally, we compared our result with the given form to find the values of $\lambda$ and $f(\theta)$. The ordered pair is $(-1, 1 + \tan \theta)$.

Final Answer The final answer is \boxed{(-1, 1 + \tan \theta)}, which corresponds to option (C).