Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.
∫ csc θ dθ

검증된 단계별 안내

Recognize that the integral involves the cosecant function, which can be tricky to integrate directly. The substitution \(z = \tan\left(\frac{\theta}{2}\right)\) is a common technique called the Weierstrass substitution, useful for trigonometric integrals.

Express \(\sin \theta\) and \(d\theta\) in terms of \(z\). Recall the half-angle formulas: \(\sin \theta = \frac{2z}{1 + z^2}\) and \(d\theta = \frac{2}{1 + z^2} dz\).

Rewrite the integral \(\int \csc \theta \, d\theta\) as \(\int \frac{1}{\sin \theta} d\theta\). Substitute the expressions for \(\sin \theta\) and \(d\theta\) in terms of \(z\) to transform the integral into one involving \(z\) only.

Simplify the resulting integral in terms of \(z\). This will typically reduce to a rational function of \(z\), which is easier to integrate using standard calculus techniques.

Integrate with respect to \(z\), then substitute back \(z = \tan\left(\frac{\theta}{2}\right)\) to express the answer in terms of \(\theta\).

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주요 개념

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Weierstrass Substitution (t = tan(θ/2))

The substitution t = tan(θ/2) transforms trigonometric integrals into rational functions of t, simplifying integration. It uses half-angle identities to rewrite sine and cosine in terms of t, making complex trigonometric integrals more manageable.

Integration of Cosecant Function

The integral of csc θ involves recognizing its standard antiderivative or transforming it using substitutions. Understanding how to rewrite csc θ in terms of sine and applying substitution techniques is essential for evaluating the integral.

Trigonometric Identities and Half-Angle Formulas

Half-angle formulas express sine and cosine in terms of tan(θ/2), enabling substitution methods. Mastery of these identities is crucial to convert the integral into a rational function and to back-substitute after integration.

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.∫ csc θ dθ