Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
32. ∫ csc²(6x) cot(6x) dx
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7. Antiderivatives & Indefinite Integrals
Integrals of Trig Functions
1:14 minutesProblem 4.R.95
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫(1 + 3 cosΘ) dΘ
Verified step by step guidance1
Step 1: Recognize that the integral is in the form ∫(1 + 3 cosΘ) dΘ, which can be split into two separate integrals: ∫1 dΘ + ∫3 cosΘ dΘ.
Step 2: Solve the first integral ∫1 dΘ. The integral of 1 with respect to Θ is simply Θ, because the derivative of Θ is 1.
Step 3: Solve the second integral ∫3 cosΘ dΘ. Use the constant multiple rule, which allows you to factor out the constant 3, resulting in 3∫cosΘ dΘ.
Step 4: Recall the integral of cosΘ with respect to Θ is sinΘ. Therefore, ∫cosΘ dΘ = sinΘ, and 3∫cosΘ dΘ = 3sinΘ.
Step 5: Combine the results from Step 2 and Step 4. The indefinite integral ∫(1 + 3 cosΘ) dΘ is Θ + 3sinΘ + C, where C is the constant of integration.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral
An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without limits of integration and includes a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation.
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Basic Integration Rules
Basic integration rules are fundamental formulas used to compute integrals. For example, the integral of a constant 'a' is 'aΘ', and the integral of cos(Θ) is sin(Θ). Understanding these rules is essential for solving integrals efficiently and accurately.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that arise in various mathematical contexts, including calculus. Their properties, such as periodicity and symmetry, play a crucial role in integration, especially when dealing with integrals involving trigonometric expressions.
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