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.
∫ (sec t + cot t)² dt

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Start by expanding the integrand \((\sec t + \cot t)^2\) using the algebraic identity \((a + b)^2 = a^2 + 2ab + b^2\). So, write the integrand as \(\sec^2 t + 2 \sec t \cot t + \cot^2 t\).

Recall the Pythagorean identities: \(\sec^2 t = 1 + \tan^2 t\) and \(\cot^2 t = \csc^2 t - 1\). Substitute these into the expression to rewrite the integrand in terms of \(\tan t\) and \(\csc t\).

Rewrite the integral as \(\int \left(1 + \tan^2 t + 2 \sec t \cot t + \csc^2 t - 1\right) dt\), which simplifies to \(\int \left(\tan^2 t + 2 \sec t \cot t + \csc^2 t\right) dt\).

Consider splitting the integral into three separate integrals: \(\int \tan^2 t \, dt + 2 \int \sec t \cot t \, dt + \int \csc^2 t \, dt\). This allows you to handle each term individually.

Evaluate each integral using known antiderivatives: \(\int \tan^2 t \, dt\) can be rewritten using \(\tan^2 t = \sec^2 t - 1\), \(\int \sec t \cot t \, dt\) can be approached by substitution, and \(\int \csc^2 t \, dt = -\cot t + C\). Combine all results to express the final integral.

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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 expressions and integrals by rewriting complex functions into more manageable forms, such as using (sec t + cot t)² = sec² t + 2 sec t cot t + cot² t.

Integration Techniques

Integration techniques include methods like substitution, integration by parts, and recognizing standard integral forms. Choosing the right technique helps to evaluate integrals efficiently, especially when dealing with trigonometric functions that can be simplified or transformed.

Derivatives and Integrals of Secant and Cotangent

Knowing the derivatives and integrals of sec t and cot t is essential. For example, the derivative of sec t is sec t tan t, and the derivative of cot t is -csc² t. These relationships help in recognizing integrable forms and applying substitution effectively.

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.∫ (sec t + cot t)² dt