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

∫ (csc² Θ + 2Θ² - 3Θ) dΘ

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Step 1: Break down the integral into separate terms. The given integral is ∫ (csc² Θ + 2Θ² - 3Θ) dΘ. Using the property of linearity of integration, rewrite it as ∫ csc² Θ dΘ + ∫ 2Θ² dΘ - ∫ 3Θ dΘ.

Step 2: Solve the first term ∫ csc² Θ dΘ. Recall that the integral of csc² Θ is a standard result: ∫ csc² Θ dΘ = -cot Θ + C₁, where C₁ is the constant of integration.

Step 3: Solve the second term ∫ 2Θ² dΘ. Use the power rule for integration: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C. Applying this rule, ∫ 2Θ² dΘ = (2Θ³)/3 + C₂, where C₂ is the constant of integration.

Step 4: Solve the third term ∫ 3Θ dΘ. Again, use the power rule for integration: ∫ x dx = (x²)/2 + C. Applying this rule, ∫ 3Θ dΘ = (3Θ²)/2 + C₃, where C₃ is the constant of integration.

Step 5: Combine the results from all terms. The indefinite integral becomes -cot Θ + (2Θ³)/3 - (3Θ²)/2 + C, where C is the overall constant of integration (combining C₁, C₂, and C₃). To check your work, differentiate the result and verify that it matches the original integrand.

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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 often referred to as antiderivation, where we seek a function F(Θ) such that F'(Θ) equals the integrand.

Basic Integration Rules

To solve indefinite integrals, one must be familiar with basic integration rules, such as the power rule, which states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. Additionally, specific functions like trigonometric functions have their own integration formulas, such as ∫csc²(Θ) dΘ = -cot(Θ) + C. Mastery of these rules is essential for effectively computing integrals.

Verification by Differentiation

After finding an indefinite integral, it is crucial to verify the result by differentiation. This involves taking the derivative of the antiderivative obtained and checking if it equals the original integrand. This step ensures that the integration process was performed correctly and helps reinforce the relationship between differentiation and integration, as they are inverse operations.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.∫ (csc² Θ + 2Θ² - 3Θ) dΘ