Indicate which formula applies to the following integral problem, then use it to evaluate 11. indicate which formula applies to the followingintegral problem. Then use it to evaluate39x²V16 – 9x2

Answered: 11. indicate which formula applies to…

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter5: Factoring Polynomials

Section5.12: Solving Equations By Factoring

Problem 52WE

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Question

Indicate which formula applies to the following integral problem, then use it to evaluate

Expert Solution

Step 1

To evaluate

$I = \int \frac{3}{9 x^{2} \sqrt{16 - 9 x^{2}}} d x$

Step 2

To solve the given integral apply trigonometric substitution

$\begin{array}{rcl} x & = & \frac{4}{3} \sin θ \\ d x & = & \frac{4}{3} \cos θ d θ \\ \sin θ & = & \frac{3}{4} x \\ θ & = & \sin^{- 1} (\frac{3}{4} x) \end{array}$

Step 3

Thus,

$\begin{array}{rcl} I & = & \int \frac{3}{9 x^{2} \sqrt{16 - 9 x^{2}}} d x \\ I & = & \frac{1}{3} \int \frac{1}{{(\frac{4}{3} \sin θ)}^{2} \sqrt{16 - 9 {(\frac{4}{3} \sin θ)}^{2}}} (\frac{4}{3} \cos θ) d θ \\ I & = & \frac{9 \times 4}{3 \times 3 \times 16} \int \frac{\cos θ}{\sin^{2} θ \sqrt{16 - 16 \sin^{2} θ}} d θ \\ I & = & \frac{1}{4 \times 4} \int \frac{\cos θ}{\sin^{2} θ \sqrt{1 - \sin^{2} θ}} d θ \\ I & = & \frac{1}{16} \int \frac{\cos θ}{\sin^{2} θ \cos θ} d θ \\ I & = & \frac{1}{16} \int c s c^{2} θ d θ \\ I & = & - \frac{1}{16} c o t θ + C \end{array}$

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