The image shows a mathematical expression for integration. The expression is:\[ \int \frac{\cos 5\beta}{\sin^2 5\beta - \sin 5\beta} \, d\beta \]This is an integral that involves trigonometric functions, specifically the cosine and sine functions, with a variable $\beta$. The expression in the denominator is a quadratic in terms of $\sin 5\beta$. Integrals of this form can often be approached using trigonometric identities or substitution techniques.

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Description: The image shows a mathematical expression for integration. The expression is:\[ \int \frac{\cos 5\beta}{\sin^2 5\beta - \sin 5\beta} \, d\beta \]This is an integral that involves trigonometric functions, specifically the cosine and sine functions, w

Consider the provided integral, ∫cos5βsin25β-sin5βdβ Find the expression for the provided integral.…

Answered: Cos 56 CUST J sin'sB-sinsB dB

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The image shows a mathematical expression for integration. The expression is:

\[ \int \frac{\cos 5\beta}{\sin^2 5\beta - \sin 5\beta} \, d\beta \]

This is an integral that involves trigonometric functions, specifically the cosine and sine functions, with a variable $\beta$. The expression in the denominator is a quadratic in terms of $\sin 5\beta$. Integrals of this form can often be approached using trigonometric identities or substitution techniques.

Expert Solution

Step 1

Consider the provided integral,

$\int \frac{\cos 5 β}{\sin^{2} 5 β - \sin 5 β} d β$

Find the expression for the provided integral.

Apply the u-substitution $u = 5 β \Rightarrow \frac{d u}{d β} = 5$ .

$\begin{array}{rcl} \int \frac{\cos 5 β}{\sin^{2} 5 β - \sin 5 β} d β & = & \int \frac{\cos u}{5 (\sin^{2} u - \sin u)} d u \\ = & \frac{1}{5} \cdot \int \frac{\cos u}{\sin^{2} u - \sin u} d u \end{array}$

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