Evaluate the integrals in Exercises 39–54.
∫ 1 / (cos θ + sin 2θ) dθ

검증된 단계별 안내

Start by rewriting the integral \( \int \frac{1}{\cos \theta + \sin 2\theta} \, d\theta \) and recall the double-angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substitute this into the integral to get \( \int \frac{1}{\cos \theta + 2 \sin \theta \cos \theta} \, d\theta \).

Factor \( \cos \theta \) from the denominator: \( \cos \theta + 2 \sin \theta \cos \theta = \cos \theta (1 + 2 \sin \theta) \). So the integral becomes \( \int \frac{1}{\cos \theta (1 + 2 \sin \theta)} \, d\theta \).

Rewrite the integral as \( \int \frac{1}{\cos \theta (1 + 2 \sin \theta)} \, d\theta = \int \frac{\sec \theta}{1 + 2 \sin \theta} \, d\theta \), since \( \sec \theta = \frac{1}{\cos \theta} \).

Use the substitution \( u = 1 + 2 \sin \theta \). Then, compute \( du = 2 \cos \theta \, d\theta \), which implies \( d\theta = \frac{du}{2 \cos \theta} \).

Rewrite the integral in terms of \( u \) and \( \theta \), and express \( \sec \theta \, d\theta \) using the substitution. This will allow you to simplify the integral and proceed with integration techniques such as partial fractions or further substitutions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are essential for simplifying expressions like cos θ + sin 2θ, where sin 2θ can be rewritten using the double-angle identity sin 2θ = 2 sin θ cos θ to facilitate integration.

Integration of Rational Trigonometric Functions

Integrating rational functions involving trigonometric expressions often requires algebraic manipulation or substitution. Recognizing patterns or rewriting the integrand in terms of a single trigonometric function can simplify the integral and make standard integration techniques applicable.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. For trigonometric integrals, substituting expressions like u = sin θ or u = cos θ can transform the integral into a more manageable form, allowing the use of basic integral formulas.

Evaluate the integrals in Exercises 39–54.∫ 1 / (cos θ + sin 2θ) dθ