use the accompanying sections of a table of integrals to evaluate the following indefinite integral. the…

use the accompanying sections of a table of integrals to evaluate the following indefinite integral. the integral may require preliminary work, such as completing the square or changing variables, before it can be found in a table.\n int\frac{4sqrt{ln^{2}x + 4}}{x}dx \nclick here to view basic integrals.\nclick here to view trigonometric integrals.

use the accompanying sections of a table of integrals to evaluate the following indefinite integral. the integral may require preliminary work, such as completing the square or changing variables, before it can be found in a table.\n int\frac{4sqrt{ln^{2}x + 4}}{x}dx \nclick here to view basic integrals.\nclick here to view trigonometric integrals.

Answer

Explanation:

Step1: Make a substitution

Let (u = \ln x), then (du=\frac{1}{x}dx). The integral (\int\frac{4\sqrt{\ln^{2}x + 4}}{x}dx) becomes (\int4\sqrt{u^{2}+4}du).

Step2: Recall integral formula

The integral formula for (\int\sqrt{a^{2}+u^{2}}du=\frac{u}{2}\sqrt{a^{2}+u^{2}}+\frac{a^{2}}{2}\ln(u + \sqrt{a^{2}+u^{2}})+C), where (a = 2) in our case. So (\int4\sqrt{u^{2}+4}du=4\left(\frac{u}{2}\sqrt{4 + u^{2}}+\frac{4}{2}\ln(u+\sqrt{4 + u^{2}})\right)+C).

Step3: Substitute back (u=\ln x)

We get (4\left(\frac{\ln x}{2}\sqrt{4+\ln^{2}x}+ 2\ln(\ln x+\sqrt{4+\ln^{2}x})\right)+C=2\ln x\sqrt{\ln^{2}x + 4}+8\ln(\ln x+\sqrt{\ln^{2}x + 4})+C).

Answer:

(2\ln x\sqrt{\ln^{2}x + 4}+8\ln(\ln x+\sqrt{\ln^{2}x + 4})+C)