Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.

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Start with the integral: \(\int \sec \theta \, d\theta\).

Multiply the integrand by \(\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}\), which is equivalent to multiplying by 1, so the integral becomes \(\int \sec \theta \cdot \frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta} \, d\theta\).

Rewrite the numerator as \(\sec \theta (\sec \theta + \tan \theta) = \sec^2 \theta + \sec \theta \tan \theta\), so the integral is now \(\int \frac{\sec^2 \theta + \sec \theta \tan \theta}{\sec \theta + \tan \theta} \, d\theta\).

Let \(u = \sec \theta + \tan \theta\). Then compute \(\frac{du}{d\theta}\) by differentiating \(u\) with respect to \(\theta\): \(\frac{du}{d\theta} = \sec \theta \tan \theta + \sec^2 \theta\).

Notice that the numerator of the integrand matches \(\frac{du}{d\theta}\), so rewrite the integral as \(\int \frac{du}{u}\), which can be integrated using the natural logarithm function.

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주요 개념

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

Integration of Trigonometric Functions

This involves techniques to find antiderivatives of trigonometric expressions. Understanding how to manipulate functions like secant and tangent is essential for simplifying integrals and applying substitution methods effectively.

Multiplying by a Conjugate Expression

Multiplying the integrand by (sec θ + tan θ) / (sec θ + tan θ) is a strategic algebraic step that simplifies the integral. This technique leverages identities to rewrite the integral in a form that is easier to integrate.

U-Substitution Method

U-substitution is a method for integrating composite functions by substituting a part of the integrand with a new variable u. It simplifies the integral by transforming it into a basic form, often turning complicated trigonometric integrals into standard ones.

Evaluate ∫ sec θ dθ by:a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.