23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ ((2 + 3 cos y)/sin² y)dy

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Step 1: Break down the integral into simpler components. Rewrite the integrand as (2/sin²y) + (3 cos y/sin²y). This allows us to handle each term separately.

Step 2: Recognize that 2/sin²y can be rewritten using a trigonometric identity. Recall that 1/sin²y = csc²y, so 2/sin²y becomes 2 csc²y.

Step 3: For the second term, 3 cos y/sin²y, rewrite it as 3 (cos y/sin²y). Notice that cos y/sin²y can be expressed as 3 cot y csc y using trigonometric identities.

Step 4: Now, the integral becomes ∫ (2 csc²y + 3 cot y csc y) dy. Split the integral into two parts: ∫ 2 csc²y dy + ∫ 3 cot y csc y dy.

Step 5: Use standard integration formulas: ∫ csc²y dy = -cot y and ∫ cot y csc y dy = -csc y. Apply these formulas to each term and combine the results, adding the constant of integration (C) at the end.

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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially in integration and differentiation. In the given integral, 'cos y' and 'sin² y' are trigonometric functions that can often be simplified or transformed using identities, aiding in the integration process.

Integration Techniques

Various techniques exist for solving integrals, including substitution, integration by parts, and partial fraction decomposition. For the integral ∫ ((2 + 3 cos y)/sin² y)dy, recognizing the structure of the integrand allows for the application of these techniques, facilitating the integration process.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.∫ ((2 + 3 cos y)/sin² y)dy