Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.3.47

Chapter 8, Problem 8.3.47

9–61. Trigonometric integrals Evaluate the following integrals.
47. ∫ (csc⁴x)/(cot²x) dx

Verified step by step guidance

Step 1: Begin by rewriting the integrand in terms of sine and cosine functions. Recall that csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x). Substitute these definitions into the integrand: ∫ (csc⁴x)/(cot²x) dx = ∫ (1/sin⁴x) / (cos²x/sin²x) dx.

Step 2: Simplify the fraction by dividing. This results in ∫ (1/sin⁴x) * (sin²x/cos²x) dx = ∫ (sin²x/sin⁴x) * (1/cos²x) dx. Simplify further to get ∫ (1/sin²x) * (1/cos²x) dx.

Step 3: Recognize that 1/sin²x is equivalent to csc²x and 1/cos²x is equivalent to sec²x. Rewrite the integrand as ∫ csc²x * sec²x dx.

Step 4: Consider substitution or trigonometric identities to simplify the integral further. For example, you might use the identity csc²x = 1 + cot²x to express the integrand in terms of cot(x), or explore substitution methods such as u = cot(x).

Step 5: After substitution or simplification, proceed to integrate the resulting expression. Remember to back-substitute if you used a substitution method, and include the constant of integration (C) in your final answer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, cosecant, and cotangent, are fundamental in calculus, particularly in integration. They relate angles to ratios of sides in right triangles and have periodic properties. Understanding their identities and relationships is crucial for simplifying integrals involving these functions.

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In this case, recognizing patterns or using trigonometric identities can simplify the integral. Mastery of these techniques allows for the effective evaluation of integrals that may not be straightforward.

Trigonometric Identities

Trigonometric identities, such as Pythagorean identities and reciprocal identities, are equations involving trigonometric functions that hold true for all values of the variables. These identities can be used to rewrite integrals in a more manageable form, facilitating easier integration. Familiarity with these identities is vital for solving integrals involving trigonometric functions.

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