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.
∫ (dθ / cos θ - 1)

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Rewrite the integral to clarify the expression: consider the integral \( \int \frac{d\theta}{\cos \theta - 1} \).

Recognize that the denominator \( \cos \theta - 1 \) can be tricky to integrate directly, so use a trigonometric identity to simplify it. Recall that \( \cos \theta - 1 = -2 \sin^2 \left( \frac{\theta}{2} \right) \).

Substitute this identity into the integral to get \( \int \frac{d\theta}{-2 \sin^2 \left( \frac{\theta}{2} \right)} = -\frac{1}{2} \int \csc^2 \left( \frac{\theta}{2} \right) d\theta \).

Make a substitution to simplify the integral: let \( u = \frac{\theta}{2} \), so \( d\theta = 2 du \). Substitute into the integral to get \( -\frac{1}{2} \int \csc^2(u) \cdot 2 du = - \int \csc^2(u) du \).

Recall the integral formula \( \int \csc^2 x \, dx = -\cot x + C \). Use this to write the integral in terms of \( u \), then substitute back \( u = \frac{\theta}{2} \) to express the answer in terms of \( \theta \).

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify integrals by transforming complex expressions into more manageable forms, such as rewriting 1/(cos θ - 1) using identities like the Pythagorean or half-angle formulas.

Substitution Method

The substitution method involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can become easier to evaluate. For example, substituting u = cos θ - 1 can transform the integral into a rational function in terms of u.

Integration of Rational Functions

Integrating rational functions involves integrating ratios of polynomials or expressions that can be manipulated into polynomial ratios. Techniques include partial fraction decomposition or algebraic manipulation, which help break down complex fractions into simpler terms that are easier to integrate.

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.∫ (dθ / cos θ - 1)