Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (sec² Θ + sec Θ tan Θ)dΘ
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
1:31 minutesProblem 4.9.53
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (6/√(4 - 4x²))dx
Verified step by step guidance1
Recognize that the integral ∫ (6/√(4 - 4x²)) dx resembles the standard form of the arcsine function: ∫ (1/√(a² - u²)) du = arcsin(u/a) + C. Here, we need to rewrite the given integral to match this form.
Factor out the constant 6 from the integral: ∫ (6/√(4 - 4x²)) dx = 6 ∫ (1/√(4 - 4x²)) dx.
Simplify the denominator √(4 - 4x²) by factoring out 4: √(4 - 4x²) = √(4(1 - x²)) = 2√(1 - x²). Substitute this back into the integral: 6 ∫ (1/√(4 - 4x²)) dx = 6 ∫ (1/(2√(1 - x²))) dx.
Simplify further by factoring out the constant 1/2: 6 ∫ (1/(2√(1 - x²))) dx = 3 ∫ (1/√(1 - x²)) dx. Now the integral matches the standard arcsine form, where u = x and a = 1.
Apply the formula for the arcsine integral: ∫ (1/√(1 - x²)) dx = arcsin(x) + C. Therefore, the solution becomes 3 * arcsin(x) + C. Finally, check your work by differentiating the result to ensure it matches the original integrand.
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1mKey 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 often referred to as antiderivation, where we seek a function F(x) such that F'(x) equals the integrand.
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Substitution Method
The substitution method is a technique used in integration to simplify the integrand by changing variables. This involves selecting a new variable, often denoted as 'u', which is a function of 'x', and rewriting the integral in terms of 'u'. This method is particularly useful when dealing with composite functions or expressions that can be simplified through a change of variables.
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Differentiation Check
After finding an indefinite integral, it is essential 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 confirms the validity of the solution.
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