Evaluate the integrals in Exercises 39–56.
52. ∫(from π/4 to π/2)cot(t)dt

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Recall the definition of the cotangent function: \(\cot(t) = \frac{\cos(t)}{\sin(t)}\).

Recognize that the integral of \(\cot(t)\) can be expressed using a standard antiderivative: \(\int \cot(t) \, dt = \ln|\sin(t)| + C\).

Set up the definite integral using the antiderivative: \(\int_{\pi/4}^{\pi/2} \cot(t) \, dt = \left[ \ln|\sin(t)| \right]_{\pi/4}^{\pi/2}\).

Evaluate the antiderivative at the upper limit \(t = \pi/2\) and the lower limit \(t = \pi/4\): calculate \(\ln|\sin(\pi/2)|\) and \(\ln|\sin(\pi/4)|\) separately.

Subtract the value of the antiderivative at the lower limit from the value at the upper limit to find the value of the definite integral: \(\ln|\sin(\pi/2)| - \ln|\sin(\pi/4)|\).

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주요 개념

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

Definition and Properties of the Cotangent Function

Cotangent, cot(t), is the ratio of cosine to sine, cot(t) = cos(t)/sin(t). Understanding its behavior and domain is essential, especially since cotangent has discontinuities where sin(t) = 0. This helps in evaluating integrals involving cotangent and recognizing potential issues with limits.

Integration of Trigonometric Functions

Integrating cotangent involves rewriting it in terms of sine and cosine or using known integral formulas. The integral of cot(t) is ln|sin(t)| + C, which is crucial for solving definite integrals involving cotangent over a given interval.

Evaluating Definite Integrals with Logarithmic Functions

After integrating, evaluating the definite integral requires substituting the limits into the antiderivative, often involving logarithmic expressions. Careful handling of absolute values and domain restrictions ensures correct evaluation of the integral's value.

Evaluate the integrals in Exercises 39–56.52. ∫(from π/4 to π/2)cot(t)dt