Textbook Question
Evaluating integrals Evaluate the following integrals.
∫ y² (3y³ + 1)⁴ dy
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
2:28 minutesProblem 4.9.49
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ sec Θ(tan Θ + sec Θ + cos Θ)dΘ
Verified step by step guidance1
Rewrite the integral to make it easier to work with. The given integral is ∫ sec(Θ)(tan(Θ) + sec(Θ) + cos(Θ)) dΘ. Distribute sec(Θ) across the terms inside the parentheses to rewrite the integral as ∫ (sec(Θ)tan(Θ) + sec²(Θ) + sec(Θ)cos(Θ)) dΘ.
Simplify the terms inside the integral. Notice that sec(Θ)cos(Θ) simplifies to 1 because sec(Θ) = 1/cos(Θ), so sec(Θ)cos(Θ) = 1. The integral now becomes ∫ (sec(Θ)tan(Θ) + sec²(Θ) + 1) dΘ.
Break the integral into separate terms for easier computation: ∫ sec(Θ)tan(Θ) dΘ + ∫ sec²(Θ) dΘ + ∫ 1 dΘ.
Evaluate each term individually: (1) For ∫ sec(Θ)tan(Θ) dΘ, recall that the derivative of sec(Θ) is sec(Θ)tan(Θ), so the integral is sec(Θ). (2) For ∫ sec²(Θ) dΘ, recall that the derivative of tan(Θ) is sec²(Θ), so the integral is tan(Θ). (3) For ∫ 1 dΘ, the integral is simply Θ.
Combine the results from the individual terms: sec(Θ) + tan(Θ) + Θ + C, where C is the constant of integration. Finally, check your work by differentiating the result to ensure it matches the original integrand.
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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 secant (sec) and tangent (tan), are fundamental in calculus, especially in integrals involving angles. The secant function is defined as the reciprocal of the cosine function, while the tangent function is the ratio of sine to cosine. Understanding their properties and relationships is crucial for solving integrals that involve these functions.
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Differentiation Check
Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the original integrand is recovered, confirming that the integration was performed accurately. It is a vital step in calculus to validate solutions and reinforce understanding of the relationship between integration and differentiation.
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