We have to evaluate the integrals: ∫dxx21-x2 Let I=∫dxx21-x2 Taking x2 common from square root,…

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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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Step 1

We have to evaluate the integrals:

$\int \frac{d x}{x^{2} \sqrt{1 - x^{2}}}$

Let $I = \int \frac{d x}{x^{2} \sqrt{1 - x^{2}}}$

Taking $x^{2}$ common from square root,

$\begin{array}{rcl} I & = & \int \frac{d x}{x^{2} \sqrt{1 - x^{2}}} \\ = & \int \frac{d x}{x^{2} (\sqrt{x^{2}}) \sqrt{\frac{1}{x^{2}} - 1}} \\ = & \int \frac{d x}{x^{2} (x) \sqrt{\frac{1}{x^{2}} - 1}} \\ = & \int \frac{d x}{x^{3} \sqrt{\frac{1}{x^{2}} - 1}} \end{array}$

Step 2

Now solving by substitution method,

let $t = \frac{1}{x^{2}} - 1$

differentiating with respect to $x$ ,

$\begin{array}{rcl} \frac{d t}{d x} & = & \frac{d (\frac{1}{x^{2}} - 1)}{d x} \\ = & \frac{d (\frac{1}{x^{2}})}{d x} - \frac{d 1}{d x} \\ = & \frac{d x^{- 2}}{d x} - 0 \\ = & (- 2) x^{- 2 - 1} \\ = & - 2 x^{- 3} \\ = & \frac{- 2}{x^{3}} \\ \frac{d t}{- 2} & = & \frac{d x}{x^{3}} \end{array}$

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