Chapter 5: Q. 80 (page 465)

Solve given definite integral.
$\int_{3}^{5} \sqrt{x^{2} - 9} d x$

Short Answer

Expert verified

$- \frac{9}{2} \cosh^{- 1} (\frac{5}{3}) + \frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{5}{3}))$

Step by step solution

Step1. Given Information

The integral is as follows.

$\int_{3}^{5} \sqrt{x^{2} - 9} d x$

The objective is to solve the integral.

Step2. Assumptions

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

Now, derivative of above equation is solved below.

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

Step3. Solution

Now, use the identity $\cosh^{2} (u) - 1 = \sinh^{2} (u) .$

So,

$\begin{aligned} \sqrt{x^{2} - 9} = \sqrt{9 \cosh^{2} (u) - 9} \\ = 3 \sqrt{\cosh^{2} (u) - 1} \\ = 3 \sqrt{\sinh^{2} (u)} \\ = 3 \sinh (u) \end{aligned}$

Step4. Solution

The integral is solved below.

$\begin{aligned} \int 9 \sinh^{2} (u) d u \\ = 9 \int \sinh^{2} (u) d u \\ = 9 \int (\frac{1}{2} \cosh (2 u) - \frac{1}{2}) d u \\ = - 9 \int \frac{1}{2} d u + \frac{9}{2} \int \cosh (2 u) d u \\ = - \frac{9}{2} u + \frac{9}{4} \sinh (2 u) \\ = - \frac{9}{2} (\cosh^{- 1} (\frac{x}{3})) + \frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{x}{3})) \end{aligned}$

Step5. Solution

The definite integral is solved below.

$\begin{aligned} \int_{3}^{5} \sqrt{x^{2} - 9} d x \\ = {[\frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{x}{3})) - \frac{9}{2} \cosh^{- 1} (\frac{x}{3})]}_{3}^{5} \\ = [\frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{5}{3})) - \frac{9}{2} \cosh^{- 1} (\frac{5}{3})] \\ - [\frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{3}{3})) - \frac{9}{2} \cosh^{- 1} (\frac{3}{3})] \\ = - \frac{9}{2} \cosh^{- 1} (\frac{5}{3}) + \frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{5}{3})) \end{aligned}$

Therefore, the value is $- \frac{9}{2} \cosh^{- 1} (\frac{5}{3}) + \frac{9}{4} \sinh (2 \cosh^{- 1} (\frac{5}{3}))$

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Solve given definite integral.
$\int_{3}^{5} \sqrt{x^{2} - 9} d x$

Short Answer

Step by step solution

Step1. Given Information

Step2. Assumptions

Step3. Solution

Step4. Solution

Step5. Solution

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Step by step solution

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One App. One Place for Learning.

Most popular questions from this chapter

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Solve given definite integral.
$\int_{3}^{5} \sqrt{x^{2} - 9} d x$