Question

Solve given definite integral.

${\int }_4^5 \frac{1}{x\sqrt{x^2+9}}dx$

Short answer

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

Step-by-step solution

Step1. Given Information

The integral is as follow.

${\int }_4^5 \frac{1}{x\sqrt{x^2+9}}dx$

The objective is to solve the integral.

Step2. Solution

The definite integral is solved below.

$\begin{array}{l}{\int }_4^5 \frac{1}{x\sqrt{x^2+9}}dx\\ ={\left[-\frac{1}{3}{\sinh }^{-1}\left(\frac{3}{x}\right)\right]}_4^5\\ =\left[-\frac{1}{3}{\sinh }^{-1}\left(\frac{3}{5}\right)\right]-\left[-\frac{1}{3}{\sinh }^{-1}\left(\frac{3}{4}\right)\right]\\ =-\frac{1}{3}{\sinh }^{-1}\left(\frac{3}{5}\right)+\frac{1}{3}{\sinh }^{-1}\left(\frac{3}{4}\right)\end{array}$

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