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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.1.14
Chapter 8, Problem 8.1.14

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (csc t sin 3t dt)

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1
Start by rewriting the integral \( \int \csc t \sin 3t \, dt \) to understand the integrand better. Recall that \( \csc t = \frac{1}{\sin t} \), so the integrand becomes \( \frac{\sin 3t}{\sin t} \).
Use the trigonometric identity for \( \sin 3t \), which is \( \sin 3t = 3 \sin t - 4 \sin^3 t \). Substitute this into the integrand to get \( \frac{3 \sin t - 4 \sin^3 t}{\sin t} \).
Simplify the fraction by dividing each term in the numerator by \( \sin t \), resulting in \( 3 - 4 \sin^2 t \). Now the integral becomes \( \int (3 - 4 \sin^2 t) \, dt \).
Recall the Pythagorean identity \( \sin^2 t = \frac{1 - \cos 2t}{2} \). Substitute this into the integral to express everything in terms of cosine: \( \int \left(3 - 4 \cdot \frac{1 - \cos 2t}{2} \right) dt \).
Simplify the integrand and then integrate term-by-term. This will involve integrating constants and cosine functions, which are straightforward to handle.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify expressions and integrals by rewriting complex trigonometric terms into more manageable forms, such as expressing sin(3t) using angle addition formulas.
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Integration Techniques

Integration techniques include methods like substitution, integration by parts, and algebraic manipulation. Choosing the right technique helps transform the integral into a simpler form that can be directly integrated, especially when dealing with products of trigonometric functions.
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Properties of Cosecant and Sine Functions

Understanding the properties of cosecant (csc t = 1/sin t) and sine functions is essential to simplify the integrand. Recognizing how these functions interact allows for algebraic simplification, such as canceling terms or rewriting the integrand to facilitate integration.
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Related Practice
Textbook Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ √(1 - (ln x)²) / (x ln x) dx

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85. Find the volume of the solid generated by revolving the region about the y-axis.

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Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

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Textbook Question

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (8x² + 8x + 2) / (4x² + 1)² dx

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Textbook Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dθ / cos θ - 1)

Textbook Question

Evaluate the integrals in Exercises 1–14.

∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)