Chapter 5: Q. 82 (page 465)

Solve given definite integral.
$\int_{1}^{2} \frac{1}{x \sqrt{9 - x^{2}}} d x$

Short Answer

Expert verified

$[[\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{2}{3}))|] - [\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{1}{3}))|)] .$

Step by step solution

Step1. Given Information

The integral is as follows.

$\int_{1}^{2} \frac{1}{x \sqrt{9 - x^{2}}} d x$

The objective is to solve the integral.

Step2. Assumptions

To solve the integral, let $x = 3 \sin (u)$

Now, derivation of the above equation is solved below

$x = 3 \sin (u) d x = 3 \cos (u) d u$

Step3. Solution

Now, use the identity $- \sin^{2} (u) + 1 = \cos^{2} (u)$

So,

$\begin{aligned} \frac{1}{x \sqrt{9 - x^{2}}} = \frac{1}{3 \sqrt{- 9 \sin^{2} (u) + 9} \sin (u)} \\ = \frac{1}{9 \sqrt{- \sin^{2} (u) + 1} \sin (u)} \\ = \frac{1}{9 \sqrt{\cos^{2} (u)} \sin (u)} \\ = \frac{1}{9 \cos (u) \sin (u)} \end{aligned}$

Step4. Solution

The integral is solved below.

$\begin{aligned} \frac{1}{3} \int \frac{1}{\sin (u)} d u \\ = \frac{1}{3} \int (\frac{1}{2 \sin (\frac{u}{2}) \cos (\frac{u}{2})}) d u \end{aligned}$

$= \frac{1}{3} \int (\frac{1}{2 \sin (v) \cos (v)}) d v [v = \frac{u}{2}, d v = \frac{d u}{2}]$

$= \frac{1}{3} \int \frac{\sec^{2} (v)}{\tan (v)} d v$

$= \frac{1}{3} \int \frac{1}{w} d w [w = \tan v, d w = \sec^{2} (v) d v]$

$\begin{aligned} = \frac{1}{3} \ln (| w |) \\ = \frac{1}{3} \ln (| \tan (v) |) = \frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{x}{3}))|) \end{aligned}$

$= \frac{1}{3} \ln (|\tan (\frac{u}{2})|)$

Step5. Solution

The definite integral is solved below.

$\int_{1}^{2} \frac{1}{x \sqrt{9 - x^{2}}} d x$

$= {[\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{x}{3}))|)]}_{1}^{2}$

$= [\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{2}{3}))|)] - [\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{1}{3}))|)]$

Therefore, the value is $[[\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{2}{3}))|] - [\frac{1}{3} \ln (|\tan (\frac{1}{2} \sin^{- 1} (\frac{1}{3}))|)] .$

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Solve given definite integral.
$\int_{1}^{2} \frac{1}{x \sqrt{9 - x^{2}}} d x$

Short Answer

Step by step solution

Step1. Given Information

Step2. Assumptions

Step3. Solution

Step4. Solution

Step5. Solution

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Solve given definite integral.∫12 1x9−x2dx

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Step by step solution

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Step2. Assumptions

Step3. Solution

Step4. Solution

Step5. Solution

One App. One Place for Learning.

Most popular questions from this chapter

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Solve given definite integral.
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