Textbook Question
Evaluating integrals Evaluate the following integrals.
∫₀^²π cos² 𝓍/6 d𝓍
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8. Definite Integrals
Substitution
1:26 minutesProblem 8.4.2
Textbook Question
2. What change of variables is suggested by an integral containing √(x² + 36)?
Verified step by step guidance1
Recognize that the integral contains the expression √(x² + 36), which suggests a trigonometric substitution. This is because the form x² + a² is commonly associated with the Pythagorean identity.
Recall the trigonometric identity: tan²(θ) + 1 = sec²(θ). This substitution is useful for integrals involving √(x² + a²). Here, a² = 36, so a = 6.
Set up the substitution: let x = 6tan(θ). This substitution simplifies x² + 36 into a trigonometric expression using the identity.
Differentiate x = 6tan(θ) to find dx: dx = 6sec²(θ)dθ. This will replace dx in the integral.
Substitute x = 6tan(θ) and dx = 6sec²(θ)dθ into the integral. The expression √(x² + 36) will simplify to 6sec(θ), making the integral easier to evaluate.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. For integrals containing terms like √(x² + a²), a common substitution is x = a tan(θ), which transforms the integral into a trigonometric form that is easier to evaluate.
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Pythagorean Identity
The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is crucial when using trigonometric substitution, as it allows us to express the square root of a sum of squares in terms of trigonometric functions, facilitating the integration process.
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Integral Evaluation
Integral evaluation is the process of finding the antiderivative of a function or calculating the area under a curve. After performing a change of variables, such as trigonometric substitution, the integral often simplifies to a standard form that can be integrated using known techniques or formulas.
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