Textbook Question
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« sinΒ² π dπ
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.5.35
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« π csc πΒ² cot πΒ² dπ
Verified step by step guidance1
Step 1: Recognize that the integral involves a composite function and trigonometric identities. The term csc(πΒ²) cot(πΒ²) suggests a potential simplification using substitution.
Step 2: Let u = πΒ². Then, differentiate u with respect to π to find du: du = 2π dπ. This substitution will simplify the integral.
Step 3: Rewrite the integral in terms of u. Substitute πΒ² with u and replace π dπ with (1/2) du. The integral becomes (1/2) β« csc(u) cot(u) du.
Step 4: Use the standard integral formula for β« csc(u) cot(u) du, which is -csc(u). This simplifies the integral to -(1/2) csc(u) + C, where C is the constant of integration.
Step 5: Substitute back u = πΒ² to express the result in terms of the original variable. The final expression becomes -(1/2) csc(πΒ²) + C.
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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 with a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral involves determining the antiderivative of the function, which can often be done using various techniques such as substitution or integration by parts.
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Change of Variables
Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex functions or when the integrand contains products or compositions of 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. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be accurate. This step is crucial in calculus as it ensures that the integration process has been performed correctly.
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