Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3/x⁴ + 2 - 3/x²)
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
3:31 minutesProblem 4.9.41
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((2 + 3 cos y)/sin² y)dy
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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