Answered: V3/2 4 dx 2 9 + 4x?

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

Chapter2: Working With Real Numbers

Section2.6: Rules For Multiplication

Problem 32WE

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Question

Integration techniques Use the methods introduced evaluate the following integrals.

Expert Solution

Step 1

Given:

$\int_{0}^{\frac{\sqrt{3}}{2}} \frac{4}{9 + 4 x^{2}} d x$

Step 2

The objective is to evaluate the given integral

Thus the given integral is

$\int_{0}^{\frac{\sqrt{3}}{2}} \frac{4}{9 + 4 x^{2}} d x$

On taking the constant out

$\int_{0}^{\frac{\sqrt{3}}{2}} \frac{4}{9 + 4 x^{2}} d x$ = $4 \int_{0}^{\frac{\sqrt{3}}{2}} \frac{1}{9 + 4 x^{2}} d x$

On applying the integral substitution

that is $x = \frac{3}{2} u$

Then,

$\int_{0}^{\frac{\sqrt{3}}{2}} \frac{4}{9 + 4 x^{2}} d x$ = $4 \int_{0}^{\frac{\sqrt{3}}{2}} \frac{1}{6 (u^{2} + 1)} d u$

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