The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (tan θ + 3 / sin θ) dθ

Verified step by step guidance

1. Rewrite the integral by separating it into two simpler integrals: \(\int \tan \theta \, d\theta + \int \frac{3}{\sin \theta} \, d\theta\).
2. Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so the first integral becomes \(\int \frac{\sin \theta}{\cos \theta} \, d\theta\).
3. For the first integral, use the substitution method: let \(u = \cos \theta\), then \(du = -\sin \theta \, d\theta\). Rewrite the integral in terms of \(u\).
4. For the second integral, recognize that \(\frac{1}{\sin \theta} = \csc \theta\), so rewrite it as \(3 \int \csc \theta \, d\theta\) and recall the standard integral formula for \(\int \csc \theta \, d\theta\).
5. Evaluate both integrals separately using the substitutions and formulas, then combine the results and add the constant of integration \(C\).

Key Concepts

Integration of Trigonometric Functions

This involves finding the antiderivative of functions involving trigonometric expressions like tan θ and sin θ. Understanding standard integrals such as ∫tan θ dθ and ∫csc θ dθ is essential for solving these integrals efficiently.

Trigonometric Identities

Trigonometric identities, such as expressing tan θ as sin θ/cos θ or rewriting terms to simpler forms, help simplify integrals. Using identities can transform complex expressions into integrable forms, making the integration process more straightforward.

Substitution Method

Substitution involves changing variables to simplify the integral, often by letting u equal a function inside the integral. This method is useful when the integral contains composite functions or when derivatives of inner functions appear, facilitating easier integration.