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dx (3 + 4x – 4x²)/2

dx (3 + 4x – 4x²)/2

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 28E
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Question
Evaluate the integral
.2 32. dx 2\3/ (3 + 4х — 4х?)/2
Transcribed Image Text:.2 32. dx 2\3/ (3 + 4х — 4х?)/2
Expert Solution
Step 1

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\int \:\frac{x^2}{\left(3+4x-4x^2\right)^{\frac{3}{2}}}dx

=\int \frac{x^2}{\left(-4x^2+4x+3\right)^{\frac{3}{2}}}dx

=\int \frac{x^2}{\left(-4\left(x^2-x\right)+3\right)^{\frac{3}{2}}}dx

add and subtract 1/4

=\int \frac{x^2}{\left(-4\left(x^2-x+\frac{1}{4}-\frac{1}{4}\right)+3\right)^{\frac{3}{2}}}dx

=\int \frac{x^2}{\left(-4\left(\left(x-\frac{1}{2}\right)^2-\frac{1}{4}\right)+3\right)^{\frac{3}{2}}}dx

=\int \frac{x^2}{\left(-4\left(x-\frac{1}{2}\right)^2+1+3\right)^{\frac{3}{2}}}dx

=\int \frac{x^2}{\left(-4\left(x-\frac{1}{2}\right)^2+4\right)^{\frac{3}{2}}}dx................(1)

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