Question

Calculate each of the definite integrals in Exercises 47–52. Some integrals require partial fractions or polynomial long division, and some do not.

${\int }_3^4\frac{4}{x^2-4}dx$

Short answer

The value is$-\ln \left(3\right)+\ln \left(5\right)$

Step-by-step solution

Step 1. Given Information:

Given definite integral ${\int }_3^4\frac{4}{x^2-4}dx$

We want to calculate each of the definite integrals.

Step 2. Calculation:

We have

$$\begin{gathered} {\int }_3^4\frac{4}{x^2-4}dx=4{\int }_3^4\frac{1}{x^2-4}dx \\ Substitutionx=2u \\ Whenx=4thenu=2andwhenx=3thenu=\frac{3}{2} \\ andalsodx=2du \\ ApplyIntegralSubstitution: \\ 4{\int }_{\frac{3}{2}}^2\frac{1}{4(u^2-1)}2du=2{\int }_{\frac{3}{2}}^2\frac{1}{(u^2-1)}du \\ =2{\int }_{\frac{3}{2}}^2\frac{1}{-(-u^2+1)}du \\ =-2{\int }_{\frac{3}{2}}^2\frac{1}{1-u^2}du \\ =-2{{\left[\frac{\ln \left|u+1\right|}{2}-\frac{\ln \left|u-1\right|}{2}\right]}^2}_{\frac{3}{2}} \\ =-\left[\ln \left|u+1\right|-\ln \left|u-1\right|\right]{{}^2}_{\frac{3}{2}} \\ =-\left[\ln \left|3\right|-\ln \left|1\right|-\ln \left|\frac{5}{2}\right|-\ln \left|\frac{1}{2}\right|\right] \\ Use\log properties: \\ =\ln \left(3\right)-\ln \left(\frac{5}{2}\right)-\ln \left(2\right) \\ =-\ln \left(3\right)+\ln \left(5\right) \end{gathered}$$