Chapter 5: Q. 81 (page 465)

Solve given definite integral.
$\int_{4}^{5} \frac{1}{x \sqrt{x^{2} + 9}} d x$

Short Answer

Expert verified

$- \frac{1}{3} \sinh^{- 1} (\frac{3}{5}) + \frac{1}{3} \sinh^{- 1} (\frac{3}{4}) .$

Step by step solution

Step1. Given Information

The integral is as follow.

$\int_{4}^{5} \frac{1}{x \sqrt{x^{2} + 9}} d x$

The objective is to solve the integral.

Step2. Solution

The definite integral is solved below.

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

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

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Solve given definite integral.
$\int_{4}^{5} \frac{1}{x \sqrt{x^{2} + 9}} d x$

Short Answer

Step by step solution

Step1. Given Information

Step2. Solution

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Solve given definite integral.
$\int_{4}^{5} \frac{1}{x \sqrt{x^{2} + 9}} d x$